The number of Conservative MPs who are prepared to take Britain out of the EU with no deal is now greater than the total number of Conservatives elected to parliament in 1997. That is not the only way that British politics has changed since then. Parties have a lot of different positions over 22 years, but the capture of the Tories by no-dealism has happened within 22 months.

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When Tony Blair’s landslide victory reduced the Tory MPs to a rump of 165, Euroscepticism meant something different. It was all about obstructing Britain’s entry into the single currency, resisting new treaties and grumbling about old ones. Only on the outermost political fringe would anyone advocate shredding the treaties, kicking away the legal and diplomatic foundations of Britain’s partnership with the continent and seeing what happened. Even three years ago, no one in the official leave campaign thought that was a good idea. Now 170 Tory MPs – more than half of the party’s parliamentary cohort – are signed up. That number, which includes many cabinet ministers, is reported to have endorsed a letter urging the prime minister to deliver the UK out of the EU promptly and regardless of the terms.

This runaway radicalisation has many causes, of which the most recent is Theresa May’s declaration of intent to stand down before Brexit’s next phase begins. The prospect of a leadership race diverts any party’s focus towards the visceral prejudices of its activist base. In the Conservative context, it generates a dark gravity that pulls everyone away from rational discussion of Britain’s strategic European interests. This does not bode well for the second round of indicative votes on possible Brexit plans in parliament.

The ferocity of Tory anti-Brussels feeling is now so intense that any soft-Brexit bloc is bound for a Labour majority

The theory is that MPs will take control of the agenda and, in a spirit of cross-party compromise, alight on a viable option that can command a Commons majority. But there is a bug in the system. A cross-party majority in parliament still needs a government to enact its choices. The current government, for all its divisions and dysfunctions, is still a Conservative one. May still clutches the seals of office. It would be tricky enough for a Tory administration to do the bidding of a coalition that included dozens of opposition MPs. But the ferocity of anti-Brussels feeling in the Conservative ranks is now so intense that any bloc in favour of softening Brexit is bound to have a Labour majority.

This was obvious from the first round of indicative votes. The “common market 2.0” plan that has the Tory MP Nick Boles as one of its sponsors won favour from 188 MPs. But only 36 of them were Conservatives. A more popular option that day was a customs union. That got 265 votes – no majority but not miles away from one. But of that 265, only 33 were Tories. Only eight Tories backed the idea of a confirmatory referendum. Those are paltry contributions to a compromise plan, compared with the 170 who are relaxed about no deal.

The indicative votes are not meant to forge lifelong political partnerships. The purpose is finding a platform that at least 320 MPs (the threshold for a working majority) can bear to be on together for long enough to get through the current crisis. It is hard for that project to work if the ruling party cannot supply enough bodies.

This is not just a flaw in the indicative votes process. It is an obstacle to the formation of any kind of national unity government – an idea that has been gaining currency in Westminster steadily over the past few weeks. It appeals mostly to moderate Tories appalled by the fanaticism of their colleagues. Sir John Major went public with his support for a grand coalition over the weekend.

But when politicians envisage bi-partisan administrations they still have to imagine a prime minister and a cabinet at the top. And since Tory MPs would be furnishing a minority of MPs – perhaps only a third – to vote for the Brexit plan that would be the unity government’s sole purpose, it isn’t obvious why a Tory should get to steer that enterprise from Downing Street. But try suggesting a national unity government under a Labour prime minister – even a figure who is respected by moderate Tories such as Yvette Cooper or Hilary Benn – and the idea quickly looks more far-fetched.

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Since averting the worst of all Brexits became a frantic race against time, the focus has been on agreeing some kind of plan. And as the Commons deadlock has tightened, the ambition has shrunk to winning at least one vote, on one day, by a majority of at least one. In reality, a workable plan means legislation that runs the full gauntlet of amendment and rejection in both chambers of parliament. It needs a majority that can endure for weeks, maybe months.

The indicative votes process is a fine symbol of the cross-party spirit in which Brexit might and should have been undertaken. It was obvious after the last election that, in a hung parliament, traditional party allegiances would need to be suspended to get a deal done. Even Julian Smith, the Tory chief whip, now accepts that the Commons arithmetic in 2017 demanded a change of course.

But party lines have proved to be one of the stickier bricks in the Jenga tower of conventional habits that passes for a constitution in this country. It is a bit late now to try to pluck them out. And while it might be possible to envisage a majority in this parliament for a sensible Brexit settlement, it is not possible to imagine Tories making up a majority of that majority. That is an immovable problem in this parliament and a reason the whole tower could easily come down.

• Rafael Behr is a Guardian columnist