Theresa May squeezed every drop of juice out of Brexit means Brexit. Now, the phrase has finally been despatched to the speechwriters’ knacker’s yard. It has served its purpose well, which was to persuade Tory leave voters – the majority of the party – that they could trust the former remainer with the Brexit cause. At Birmingham this week they decided that they can.

But as one banal catchphrase exits stage left, another enters stage right to fill its place. The new May cliche is that she is not going to provide a running commentary on the Brexit negotiations. She said it to Andrew Marr in an interview at the start of the conference on Sunday, and on Wednesday in her main address. She will doubtless say it again when MPs return to parliament next week.

At first hearing it sounds sensible and grown up. It is meant to. Brexit is indisputably serious statecraft. No task on May’s agenda is bigger, more complex or more freighted. What could be more reasonable than to let May get on with the job undisturbed rather than have journalists, lobbyists and MPs constantly trying to distract or misrepresent her?

Yet the more you think about it, the clearer it is that it’s not very reasonable at all and that it should be challenged. What May is really saying is she does not want to be held to detailed account about Europe until the next election. She wants to spend the next three years deciding what combination of rules and deals about the movement of capital, goods and labour with Europe will enable her to sell the reality of Brexit to the Labour and Ukip voters she is courting so skilfully. And she clearly wants to do this in secret.

This is how May works. It’s why David Cameron’s aides dubbed her “submarine May”. By saying she is not going to keep up a running commentary, submarine May is, in effect, preparing to put to sea once more, and for as prolonged a stay on the ocean floor as she can manage. She even said this week that it was not in the national interest for her to talk about the talks. But this is not the national interest. It is self-interest masquerading as the national interest. Once a home secretary always a home secretary.

Liam Fox
‘Liam Fox always has a whiff of PG Wodehouse’s absurd man of destiny Roderick Spode about him.’ Photograph: Toby Melville/Reuters

May’s attempt to go off the radar is a huge challenge to the resourcefulness of domestic politics. Before 23 June, Brexit was one question, in or out. Since 23 June it has become a host of hugely important consequential questions about access, rights and rules, all of which involve big principles and have huge economic, political and cultural implications. To talk of this as “hard” or “soft” Brexit is simply banal.

May knows this. In a little-noticed part of her Wednesday speech she said that the negotiations with the EU would involve “give and take”. It was one of the understatements of the year. So far she has been notably skilful in finessing the reality of Brexit to the satisfaction of Britain’s tabloids. But the tough choices involved in a give-and-take deal are exactly why she does not wish to draw much attention to what she, her ministers and her EU counterparts are discussing until a deal is struck.

Sometimes, though, the egos and contradictions cannot be contained. This week, with all ministers on best behaviour, there have been two explosions. In one, Downing Street had to slap down a David Davis aide’s claim that Britain will never again make a contribution to the EU budget, something on which May has been notably and perhaps significantly silent.

Meanwhile the international trade secretary Liam Fox, who always has a whiff of PG Wodehouse’s absurd man of destiny, Roderick Spode, about him, is reportedly at loggerheads with the chancellor Philip Hammond, who has submarine qualities himself. Fox wants the UK to quit the European customs union. Hammond doesn’t. It’s a really big issue. It is surely in the national interest to have it out in public, not to batten down the hatches and refuse to talk about it.

In the end, all this is about the role of parliament after the referendum. We can be confident that May never forgets the House of Commons has a Tory working majority of only 16. But she also knows that the Commons is, or was, overwhelmingly opposed to Brexit. Before the referendum, there were 475 MPs in favour of remain, 158 in favour of leave. Tory MPs were 185 to 138 in favour of remain.

Clearly, that would not be the case now, in part for honourable democratic reasons. The people have spoken, and MPs must respect that. But that doesn’t mean MPs have to say nothing at all, to ask no questions, to challenge no minister or never use their votes to shape the withdrawal process. Those 475 remain MPs ought to accept the overarching Brexit vote, but otherwise make trouble. They have a right and a responsibility to ensure the outcome is the least bad it can be.

David Cameron, defeated by the Brexit vote, prepares to exit the Commons. But there are enough backbenchers of all persuasion to ensure the leave camp does not have it all its own way.

How best to make their numbers count in a way that does not appear to challenge the referendum result? It is hard to think of a more important question facing parliament today, both in terms of public policy and of functioning democracy. On a formal level, unfortunately, it is hard to see the opposition parties making sufficient common cause on a strategy that would persuade Conservative Brexit opponents to take part. Labour is distracted. The SNP is playing its own game. The Lib Dems are tiny.

The answer lies on the backbenches. On Labour and Tory sides, as well as in the SNP and the smaller parties, there are large numbers of serious and experienced pro-Europeans with different concerns. Is it so unthinkable that people like Ed Miliband, Angus Robertson, Nick Clegg, George Osborne and Caroline Lucas could combine to ensure parliament’s majority does more to shape the Brexit process?

So far, the running on Brexit terms has been made on the Tory right and by the triumphalist anti-European press. MPs need to ensure that May takes the pro-Europeans at least as much into account as she does the anti-Europeans. The running commentary she is so reluctant to give is the very least they should be seeking.