SO is this it? Are we really about to see Theresa May compromise?

It seems so unlikely – like hearing Donald Trump say sorry, or Vladimir Putin share his Victoria sponge recipe – that it has been hard to compute. Mrs May has stuck rigidly to the same course for so long that an Edinburgh tram looks vacillating by comparison.

Naturally she’s approaching compromise in her own inimitable style. In her little iMovie to the nation on Sunday, the steeliness was back. She sounded intransigent in her flexibility; hell-bent on seeing both sides.

Of course, her video message was not just about explaining why she’s pursuing cross-party talks with Labour. It was also about putting the thumbscrews on Tory MPs who might still be persuaded to vote for her deal.

And there is still no guarantee that a meaningful cross-party deal can be done. Labour sources have echoed Nicola Sturgeon’s complaint that so far, Mrs May’s idea of compromise has involved sticking to her position and inviting others to agree with her. In any case, it’s hard to see how Labour can sign up to a deal her successor will be not be bound by.

But at some point, something has to give. People are going to have to shift their positions if the uncertainty is to end. No one is saying that will be easy. If the Prime Minister is talked into backing a permanent customs union, perhaps allied to single market membership – particularly if her deal were contingent on a confirmatory vote – there would be apoplexy on the Tory right. What would happen to Mark Francois is anyone’s guess. He’s already called for an informal vote of no confidence in Mrs May. He seems to see himself as the avenging avatar in a video game, tweeting portentously that “after hubris, comes nemesis”. Oh dear. I fear he may try to settle Brexit by arm-wrestling Tom Watson. And that’s just Mr Francois. John Redwood could malfunction and stall. Steve Baker could cry.

But Remainers would be upset too. If this mess ever gets sorted out, most MPs will probably not get their first choice. Instead, they will have to back a compromise candidate, a middle-of-the-roader, the Phillip Schofield of Brexits. You can’t please all of the people all of the time, and thanks to the hopelessly vague Brexit proposition, which meant so many different things to different people, the Brexit compromise if and when it finally lands seems destined to displease pretty much everybody for good.

But at least with a soft Brexit, businesses, NHS managers, the security services and huge numbers of voters would breathe a sigh of relief that we were finally making some progress without torpedoing the economy, our wellbeing and security in the process.

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The need for a solution could hardly be more pressing, but time and again, politicians in the two main UK parties (with some heroic backbench exceptions) have veered away from finding common ground.

Narrow party interest is one reason why. Another is that in Mrs May and Jeremy Corbyn, we have two figures who are both peculiarly unbending and lacking in leadership qualities.

A third is that both the main parties have become dominated by the ideologies of their more extreme wings.

But overarching it all, we have Westminster’s dysfunctional political culture. The UK’s voting system allows parties with a minority of votes but a majority of seats to impose their will on us all. Hung parliaments are rare. Over many decades, this has created a climate in which a premier or Cabinet minister who seeks to open dialogue with opposition parties is seen as weak. Politicians are treated like Peter denying Christ if they diverge from their manifestos. Theresa May sought right from the beginning to exclude not only opposition parties from her Brexit deliberations, but Parliament as a whole.

It was exasperating to watch, but unsurprising. This behaviour was entirely in keeping with the winner-takes-all tradition of UK politics. Having traded in her majority for a hung parliament in 2017, and with rebel Tories refusing to toe the line, she should have changed tack and understood the need for compromise. But she didn’t, and that failure has heavily contributed to the current chaos.

Westminster is not unique in breeding politicians who want a free rein to pursue their own agendas. But politicians in some countries are better at compromising because they are forced to. They operate in parliaments elected by more proportional voting systems which require parties to form alliances. Heck, Holyrood operated with a stable Lab-Lib coalition for eight years and has coped admirably under two separate SNP minority administrations due to deal-making with the Greens and even the Tories. At council level, there have been all manner of cross-party deals in Scotland, which, unlike England, has PR for local government. Yet the typical refrain among defenders of the Westminster system, is that governments that rely on consensus are “weak”. Well, it’s not Holyrood that looks weak now.

Even if you allow that the circumstances would have been iniquitous for any prime minister tasked with delivering Brexit, Mrs May has played her hand poorly. She boxed herself in with her own red lines.

There are workable alternatives. Common Market 2.0 – the jargon for single market membership plus a customs “arrangement” – might do well in any future series of indicative votes. It would mean accepting a form of free movement (cue howls of anguish from the ERG) but would leave the UK free to sign trade deals, eliminate the need for the Irish backstop and allow for exit from the Common Fisheries and Agricultural Policies. The locus of a possible compromise is emerging around this, or a customs union and a second referendum.

If there is a compromise deal, it might prove to be the first of many. It’s starting to look as if more fragmented party support will be the domestic political legacy of Brexit, with polls showing an upturn of support for smaller parties and more than one new party emerging. Brexit itself, it is increasingly clear, will be a process that could last years.

Westminster must change. Whether or not Scotland stays, the UK Parliament and voting system must evolve. And when it does, the Westminster handbook will need a new chapter on the art of compromise.