The 20-year-old England C midfielder joined the Cod Army on a two-year contract as a free agent this week.
Fylde say Charles signed a two-year deal with them in September 2015, but the Football Association have no record of the contract being processed.
The former Blackpool trainee was therefore free to move to any club.
"It really sticks in the back of your throat when you're suddenly informed that Fleetwood have signed him," chairman David Haythornthwaite told BBC Radio Lancashire.
"I spoke to [chief executive] Steve Curwood and [chairman] Andy Pilley and told them what I thought of the situation and it was pretty poor on their behalf to do what they have done.
"The best player doesn't suddenly become available for no reason and obviously the agent had told them about the situation and they have taken the decision."
Despite their frustration at Charles moving to another club, Haythornthwaite conceded they could have checked before this situation arose.
"Our secretary sent off the paperwork in good faith - he sent all the paperwork for the past four or five years and it's always got there - and in this case it hasn't," he added.
"We've only got the FA's word that they didn't get it and I've got Martin's word that he did send it.
"What we didn't do was to check that they'd received it because they should always send us a copy of that receipt, so we're totally at fault in that respect."
Haythornthwaite confirmed they would be looking for compensation from League One Fleetwood.
BBC Radio Lancashire contacted Fleetwood to respond to these allegations but they declined to comment.
AFC Fylde have criticised Fleetwood Town for signing Dion Charles as they believe he was still contracted to the National League North side.