user9082: Suppose that, in an amateur rapid play tournament (where supervision is inappropriate), the following position appears:

[fen "5K2/8/1q6/1N1k4/8/8/8/2R1R3 w - - 1 1"]

1. Re5+? Qc5+? {illegal} 2. Rxc5#

The players do not agree on the result of the game. White claims he won in virtue of the rule:

5.1.a.
The game is won by the player who has checkmated his opponent’s king. This immediately ends the game, provided that the move producing the checkmate position was in accordance with Article 3 and Articles 4.2 – 4.7.

Black claims the game is a draw in virtue of the rule:

Appendix A.4.d.
If the arbiter observes both kings are in check, or a pawn on the rank furthest from its starting position, he shall wait until the next move is completed. Then, if the illegal position is still on the board, he shall declare the game drawn.

Remark that the position after 2.Rxc5# is indeed illegal in virtue of the rule:

3.10.c.
A position is illegal when it cannot have been reached by any series of legal moves.

The referee remarks that there is a discussion about a game, moves to this table and listens to both claims. What is the correct decision?
A related question: suppose that the referee remarks the illegal position after 1.... Qc5+, and stays near this table, and assists to the move 2.Rxc5#  What is the correct decision in this case?

user9082: In my personal opinion, the answers which follow from rules seem to be: White wins in the first case (by checkmate), draw in the second case (the situation of Appendix A.4.d happened before checkmate).  Although, this seems a bit illogical since the only difference between the two situations is the time when the referee was present near the table.

user4418: IA Petr Harasimovic's opinion notwithstanding the game is drawn. Winning the game in an illegal position is a nonsense (barring serious misconduct or a ringing mobile phone). The problem is that the English (authoratative) version of the rules is ambiguously worded - 

Then, if the illegal position is still on the board, he shall declare
  the game drawn.

Thank you, @AlexandreAksenov, for the link to the authorized French translation of the rules. They remove the ambiguity with a slight mistranslation, which I suspect may be deliberate -

si la position sur l’échiquier est toujours illégale, il déclarera la
  partie nulle

which translates back into English as "If the position on the board remains illegal, he shall declare the game drawn."
The British Chess Arbiters' Association has produced a version of the rules consisting of the FIDE rules verbatim with their own added comments to clarify points like these. Here is the relevant part of their clarification for A4.d:

Consider the following situations:  ... (b)  The arbiter arrives at
  the board.  One king is already in check.  The player completes a move
  checking the other king.  The arbiter must wait until the next move is
  completed.  If the position is still illegal he will declare the game
  drawn.

(My italics)
This is eerily similar to the French. Perhaps no coincidence that the CAA has a French version (along with German and Russian) which looks like it was taken from the French Chess association website.
For completeness sake it is probably worth noting that the FIDE Arbiters' Association also has its own handbook! It has no clarifying comments for A4.d. 
It also worth stressing that the FAA, CAA handbooks and the various translations by the different federations remain subservient to the original FIDE Laws of Chess as published on the FIDE website. The same applies to my opinions on A4.d expressed here and to the conflicting opinions of IA Petr Harasimovic. Nevertheless I suspect that the vast majority of FIDE arbiters would declare the game in question drawn.

user9030: Laws of Chess: For competitions starting on or after 1 July 2014

Article 7: Irregularities
  7.1   
If an irregularity occurs and the pieces have to be restored to a
  previous position, the arbiter shall use his best judgement to
  determine the times to be shown on the chessclock. This includes the
  right not to change the clock times. He shall also, if necessary,
  adjust the clock’s move-counter.
  7.2   
If during agame it is found that the initial position of the pieces was incorrect, the game shall be cancelled and a new game shall
  be played.
     If during a game it is found that the chessboard has been placed contrary to Article 2.1, the game shall continue but the position
  reached must be transferred to a correctly placed chessboard.
7.3   
If a game has begun with colours reversed then it shall continue,
  unless the arbiter rules otherwise.
  7.4   
If a player displaces one or more pieces, he shall re-establish the
  correct position in his own time. If necessary, either the player or
  his opponent shall stop the chessclock and ask for the arbiter’s
  assistance. The arbiter may penalise the player who displaced the
  pieces.
  7.5   
If during a game it is found that an illegal move has been completed, the position immediately before the irregularity shall be
  reinstated. If the position immediately before the irregularity cannot
  be determined, the game shall continue from the last identifiable
  position prior to the irregularity. Articles 4.3 and 4.7 apply to the
  move replacing the illegal move. The game shall then continue from
  this reinstated position.
     If the player has moved a pawn to the furthest distant rank, pressed the clock, but not replaced the pawn with a new piece, the
  move is illegal. The pawn shall be replaced by a queen of the same
  colour as the pawn.
      After the action taken under Article 7.5.a, for the first completed illegal move by a player the arbiter shall give two minutes
  extra time to his opponent; for the second completed illegal move by
  the same player the arbiter shall declare the game lost by this
  player. However, the game is drawn if the position is such that the
  opponent cannot checkmate the player’s king by any possible series of
  legal moves.
7.6   
If, during a game, it is found that any piece has been displaced from
  its correct square the position before the irregularity shall be
  reinstated. If the position immediately before the irregularity cannot
  be determined, the game shall continue from the last identifiable
  position prior to the irregularity. The game shall then continue from
  this reinstated position.

So I think, game returns to last position where no illegal move committed.
5.1.a is not applicable here, because 3.10.c violated, and satisfying Article 3 is a prerequisite for 5.1.a.
3.10.c

A position is illegal when it cannot have been reached by any series of legal moves.


user8844: The mate stands and white wins. Although the position both before and after the mating move is illegal the mating move itself is legal therefore article 5.1.a is satisfied. 
If an arbiter observes black playing 1...Qc5 then they declare the game lost for black for playing an illegal move. If they do not see the move but they observe the position after 1...Qc5 they will wait for white's move. Since white played 2.Rxc5 both kings are no longer in check hence article A.4.d does not trigger. In fact the purpose of the arbiter having to wait one move is to give the player a chance to correct the position. As it happens in this example the final position is still illegal but for a reason not covered by article A.4.d.

9082: <p>Suppose that, in an amateur rapid play tournament (where supervision is inappropriate), the following position appears:</p>
<hr />
<pre><code>[fen &quot;5K2/8/1q6/1N1k4/8/8/8/2R1R3 w - - 1 1&quot;]

1. Re5+? Qc5+? {illegal} 2. Rxc5#
</code></pre>
<p>The players do not agree on the result of the game. White claims he won in virtue of the rule:</p>
<blockquote>
<p>5.1.a.</p>
<p>The game is won by the player who has checkmated his opponent’s king. This immediately ends the game, provided that the move producing the checkmate position was in accordance with Article 3 and Articles 4.2 – 4.7.</p>
</blockquote>
<p>Black claims the game is a draw in virtue of the rule:</p>
<blockquote>
<p>Appendix A.4.d.</p>
<p>If the arbiter observes both kings are in check, or a pawn on the rank furthest from its starting position, he shall wait until the next move is completed. Then, if the illegal position is still on the board, he shall declare the game drawn.</p>
</blockquote>
<p>Remark that the position after 2.Rxc5# is indeed illegal in virtue of the rule:</p>
<blockquote>
<p>3.10.c.</p>
<p>A position is illegal when it cannot have been reached by any series of legal moves.</p>
</blockquote>
<p>The referee remarks that there is a discussion about a game, moves to this table and listens to both claims. What is the correct decision?</p>
<p>A related question: suppose that the referee remarks the illegal position after 1.... Qc5+, and stays near this table, and assists to the move 2.Rxc5#  What is the correct decision in this case?</p>
 9082: <p>In my personal opinion, the answers which follow from rules seem to be: White wins in the first case (by checkmate), draw in the second case (the situation of Appendix A.4.d happened before checkmate).  Although, this seems a bit illogical since the only difference between the two situations is the time when the referee was present near the table.</p>
 4418: <p>IA Petr Harasimovic's opinion notwithstanding the game is drawn. Winning the game in an illegal position is a nonsense (barring serious misconduct or a ringing mobile phone). The problem is that the English (authoratative) version of the rules is ambiguously worded - </p>

<blockquote>
  <p>Then, if <em>the</em> illegal position is still on the board, he shall declare
  the game drawn.</p>
</blockquote>

<p>Thank you, @AlexandreAksenov, for the link to the authorized French translation of the rules. They remove the ambiguity with a slight mistranslation, which I suspect may be deliberate -</p>

<blockquote>
  <p>si la position sur l’échiquier est <em>toujours</em> illégale, il déclarera la
  partie nulle</p>
</blockquote>

<p>which translates back into English as "If the position on the board <em>remains</em> illegal, he shall declare the game drawn."</p>

<p>The British Chess Arbiters' Association has produced a <a href="http://www.chessarbitersassociation.co.uk/FIDE_LAWS_OF_CHESS_2014_with_comments.pdf" rel="nofollow">version of the rules</a> consisting of the FIDE rules verbatim with their own added comments to clarify points like these. Here is the relevant part of their clarification for A4.d:</p>

<blockquote>
  <p>Consider the following situations:  ... (b)  The arbiter arrives at
  the board.  One king is already in check.  The player completes a move
  checking the other king.  The arbiter must wait until the next move is
  completed.  If the position is <em>still</em> illegal he will declare the game
  drawn.</p>
</blockquote>

<p>(My italics)
This is eerily similar to the French. Perhaps no coincidence that <a href="http://www.chessarbitersassociation.co.uk/law_comparison.pdf" rel="nofollow">the CAA has a French version</a> (along with German and Russian) which looks like it was taken from the French Chess association website.</p>

<p>For completeness sake it is probably worth noting that the FIDE Arbiters' Association also has <em>its</em> <a href="http://arbiters.fide.com/images/stories/downloads/2014/Arbiters-Manual-2014.pdf" rel="nofollow">own handbook</a>! It has no clarifying comments for A4.d. </p>

<p>It also worth stressing that the FAA, CAA handbooks and the various translations by the different federations remain subservient to the original FIDE Laws of Chess as published on the FIDE website. The same applies to my opinions on A4.d expressed here and to the conflicting opinions of IA Petr Harasimovic. Nevertheless I suspect that the vast majority of FIDE arbiters would declare the game in question drawn.</p>
 9030: <p><a href="https://www.fide.com/fide/handbook.html?id=171&amp;view=article" rel="nofollow">Laws of Chess: For competitions starting on or after 1 July 2014</a></p>

<blockquote>
  <p>Article 7: Irregularities
  7.1   </p>
  
  <p>If an irregularity occurs and the pieces have to be restored to a
  previous position, the arbiter shall use his best judgement to
  determine the times to be shown on the chessclock. This includes the
  right not to change the clock times. He shall also, if necessary,
  adjust the clock’s move-counter.
  7.2   </p>
  
  <p>If during agame it is found that the initial position of the pieces was incorrect, the game shall be cancelled and a new game shall
  be played.
     If during a game it is found that the chessboard has been placed contrary to Article 2.1, the game shall continue but the position
  reached must be transferred to a correctly placed chessboard.</p>
  
  <p>7.3   </p>
  
  <p>If a game has begun with colours reversed then it shall continue,
  unless the arbiter rules otherwise.
  7.4   </p>
  
  <p>If a player displaces one or more pieces, he shall re-establish the
  correct position in his own time. If necessary, either the player or
  his opponent shall stop the chessclock and ask for the arbiter’s
  assistance. The arbiter may penalise the player who displaced the
  pieces.
  7.5   </p>
  
  <p><strong>If during a game it is found that an illegal move has been completed, the position immediately before the irregularity shall be
  reinstated. If the position immediately before the irregularity cannot
  be determined, the game shall continue from the last identifiable
  position prior to the irregularity.</strong> Articles 4.3 and 4.7 apply to the
  move replacing the illegal move. The game shall then continue from
  this reinstated position.
     If the player has moved a pawn to the furthest distant rank, pressed the clock, but not replaced the pawn with a new piece, the
  move is illegal. The pawn shall be replaced by a queen of the same
  colour as the pawn.
      After the action taken under Article 7.5.a, for the first completed illegal move by a player the arbiter shall give two minutes
  extra time to his opponent; for the second completed illegal move by
  the same player the arbiter shall declare the game lost by this
  player. However, the game is drawn if the position is such that the
  opponent cannot checkmate the player’s king by any possible series of
  legal moves.</p>
  
  <p>7.6   </p>
  
  <p>If, during a game, it is found that any piece has been displaced from
  its correct square the position before the irregularity shall be
  reinstated. If the position immediately before the irregularity cannot
  be determined, the game shall continue from the last identifiable
  position prior to the irregularity. The game shall then continue from
  this reinstated position.</p>
</blockquote>

<p>So I think, game returns to last position where no illegal move committed.
5.1.a is not applicable here, because 3.10.c violated, and satisfying Article 3 is a prerequisite for 5.1.a.</p>

<p>3.10.c</p>

<blockquote>
  <p>A position is illegal when it cannot have been reached by any series of legal moves.</p>
</blockquote>
 8844: <p>The mate stands and white wins. Although the position both before and after the mating move is illegal the mating move itself is legal therefore article 5.1.a is satisfied. </p>

<p>If an arbiter observes black playing 1...Qc5 then they declare the game lost for black for playing an illegal move. If they do not see the move but they observe the position after 1...Qc5 they will wait for white's move. Since white played 2.Rxc5 both kings are no longer in check hence article A.4.d does not trigger. In fact the purpose of the arbiter having to wait one move is to give the player a chance to correct the position. As it happens in this example the final position is still illegal but for a reason not covered by article A.4.d.</p>
