MPs can always tell you the exact size of their majority. The smaller the number, the deeper it is engraved in memory. Thanks to a handy tool on the parliamentary petitions website, they can now compare that number to the volume of people in their constituencies calling for the revocation of article 50 to keep Britain in the EU. When the signatures climbed into millions, they all started checking.

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They query the total, mutter about duplicate emails and say it is a poor measure of public opinion. But still they look and do the maths: current majority minus article 50 revokers. The results are not pretty.

If it achieves nothing else, that petition has literally put remain voters back on the map and reminded many Tories why they really do not want a general election. It isn’t just because thousands of their constituents hate Brexit. They cannot imagine defending the government’s handling of the process to those who voted leave or inviting them to renew Theresa May’s contract as prime minister.

The Tories’ only comfort is that many people who don’t want May in Downing Street want Jeremy Corbyn there even less. In 2017, the opposition played the angles on Europe quite cleverly. Labour promised leavers that it would honour the referendum result, while scooping up votes from remainers whose priority was denying May a majority. But nearly two years trying to face both ways on Brexit have taken their toll. Cavilling ambiguity is never a good look for a politician, least of all Corbyn, whose brand is built on plain speaking. Now leavers are suspicious of Labour’s Europhile agitation for a second referendum and remainers are frustrated with the leader’s Eurosceptic distaste for the idea.

It is possible that raw anger with the government’s incompetence and Tory callousness are sufficient to drive a Labour surge in an election. The party’s strategists seem to be relying on that dynamic and Corbyn has a way of rediscovering form on the campaign trail. But it is also possible that 2017 was the last hurrah for English pendulum politics; that a backlash against one of two parties no longer translates into an upswing for its rival. (That mechanism stopped working years ago in Scotland.)

There is evidence that remain and leave are now more compelling drivers of political identity than party allegiance. In research published last autumn, Prof John Curtice, a leading authority on voter behaviour, noted that Brexit had not only given new shape to the electorate but also added new fire. It has stirred up “passion of which … voters had long since seemed incapable”.

It is unclear how those passions will express themselves in polling booths. The qualities of being a leaver or a remainer are unmoored from the mostly dull questions of EU institutional relations. They serve instead as code for positions in a culture war, conveying rival value judgments about the kind of country Britain once was, is now and should be. It is a faultline that cuts through the middle of parties. It separates activists from constituents in seats they are trying to defend. Labour remainers canvassing in Brexit-backing Ashfield and Tory leavers door-knocking in pro-European Putney face similar barriers to communication. It is hard to see parliament finding a compromise that will tear those down.

Indicative votes in the Commons on Wednesday will allow MPs to express technical preferences – a customs union, the single market, both, neither. Those questions need answers, but the problems of Brexit have mutated beyond the point where those answers are enough. Perhaps there was no deal that could resolve them. It certainly became harder after May had made her “citizens of nowhere” speech and Tory ultras decided that a soft Brexit would be treasonous.

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This puts parliament in an impossible position. MPs are pulled in different directions by colleagues, whips, activists, constituents and their consciences. Many think Brexit is a mistake but fear a backlash if it is aborted. They hate May’s deal, but can’t stomach another referendum. They feel the whole body politic haemorrhaging authority every day that the impasse goes on. May and Corbyn have a common character defect that inhibits engagement with people who disagree with them. MPs have mustered transient cross-party coalitions, eking out majorities on points of Brexit process. But legislating for any long-term settlement would require prolonged suspension of the usual whipping discipline. It would mean dozens of MPs abandoning any pretence of following their party leaders, not just in a single vote but over weeks. There isn’t much sign of that.

It is easy enough from the outside to call on MPs put the needs of country ahead of party. The distinction is less obvious to politicians who believe the country is, by definition, served by having their party in power. Then there are personal factors militating against political apostasy – fear of letting down colleagues, severing friendships, losing the job that pays the bills.Brexit has unleashed powerful forces against England’s two-party system, but inertia and tradition put up formidable resistance, as the breakaway Independent Group of MPs is discovering. Individual defections sting the old parties, but unless they become a swarm, it is easy enough to swat them aside. The immediate problem for Labour and the Conservatives is that Brexit has changed the political map, exposing divisions in the country that won’t align with old party templates.

May and Corbyn naturally try to manage the situation through the party structures they can just about control, but the crisis won’t be contained in those vessels. It keeps spilling out from the sides. It is washing away the rules.

That process is driving speculation about a general election in the coming months, not because MPs want one, but because it is the thing that Westminster knows how to do when nothing else is working. It is the hard reboot when the screen has frozen. You can always try turning it off and turning it back on again; see if that helps, although I doubt it. The Brexit virus is deep in the system. Dissolving parliament won’t fix the problem unless some party lines are also dissolved.

• Rafael Behr is a Guardian columnist