Brexit is a reminder that two connected things need to happen in order for a political compromise to stick. First, partisan leaders must decide that it is better than any other achievable outcome. Second, the public must also converge to give it their backing. One movement towards the treaty table without the other doesn’t cut it.

The Northern Ireland peace process was a classic example of this dual process. It took many years before mutually antagonistic political leaders decided that a peace deal was better than an unwinnable war. But it took even longer for the two communities in Northern Ireland to begin to make the same move. The peace process only worked because the political deal was followed by referendums that endorsed it.

Yet in comparison with Brexit it now looks relatively straightforward. This is partly because almost all the thinking about how to settle the Brexit argument focuses on Westminster. Seven days ago, in the wake of Theresa May’s drubbing over her deal, I wrote that it was now time for her to compromise on Brexit. I still think that’s true. But it skates over the question of whether the British people would see such a move as acceptable, and what might follow if they did not.

Finding a compromise is formidably difficult, even in Westminster. The reasons are familiar. The two main leaders are at odds with their own parties. Parliament’s law-making procedures make it hard to intervene. The tribalism of politics makes cooperation fragile. There may be a majority against no deal, but there isn’t a majority – yet – in favour of anything.

Parliament is getting heat for this failure, and the chorus is likely to grow louder next week. Yet broader opinion is divided too. There is almost no consensus among voters about what should happen now. Though MPs are often lambasted for being out of touch, the uncertainties at Westminster reflect those of the public.

At a conference in London this week organised by The UK in a Changing Europe, called Brexit and Public Opinion 2019, it became clear how deep this problem has become. In his keynote address, the political scientist John Curtice argued that the practical problem with Brexit is that there is very little middle ground on which the two sides can make trades. If you are looking for a Brexit outcome that satisfies the two key tests – first, that it should have overall majority support, and second, that it should have majority support on both sides – then the search so far has been unavailing.

May regards her own deal, not entirely unreasonably, as a compromise already. But it is not just the Commons that dislikes her deal. Twice as many voters oppose it as support it, though often for wildly different reasons. As Curtice put it this week, May has succeeded in uniting a divided country in opposition to the agreement she has made with the European Union.

Theresa May leaving 10 Downing Street. Photograph: Xinhua/Rex Features

However, none of the other alternatives is popular either. Most remainers continue to support remaining, hence their backing for a second referendum. Most leavers, though, still want to leave, with the largest group of them wanting to leave even without a deal. Compromise efforts soon run into the fact that this is a party divide too. Labour is an overwhelmingly remain party. The Tories are equally emphatically for leave. Neither side is keen enough on the options in between.

It is true that about a third of remainers say they could support a Norway-plus solution as a second-best – but leavers see this as very much a remainers’ compromise, because it accepts EU rules and regulations. For a proportion of leavers, a renegotiation is seen as the second-best option – but remainers are suspicious of that; they see going back to Brussels as a hard-Brexit ploy. The much-touted second referendum option falls into the same divide. With opinion among Tory voters now so emphatically in the leave camp, there is little incentive for May to move in the direction of a softer Brexit, let alone a second vote.

Curtice’s view is not unchallenged. In a Bloomberg article today, Matt Singh of Number Cruncher Politics argues, on the basis of NCP’s own polling, that May’s deal is in fact the closest thing to a consensus among public opinion. At first sight, this is a weird claim, since May’s deal is far less popular in Singh’s poll than either remaining in the EU or leaving with no deal. However, when people are asked if these other outcomes might be personally acceptable to them, May’s deal vaults into a narrow lead. This deal is “unloved, but it’s the closest thing to a compromise that most could live with,” Singh argues.

He could be right, if Westminster opinion this week is any guide. The determination of MPs like Yvette Cooper and Nick Boles to prevent no deal may now be alarming hard-Brexit Tories who fear the process will be delayed and softened. Some of them may decide they could live with a form of May’s deal after all. If she is able to cobble together a majority, it is possible that public opinion, exhausted by Brexit, might also fall into line.

It would not, though, be much of a compromise. The most obvious accommodation on Brexit – a soft exit along Norway lines – might have succeeded if David Cameron had decided to stay, if May had embraced it early on, or if Jeremy Corbyn had been a different kind of Labour leader. But those chances have come and gone. One of the striking aspects of the Brexit standoff since 2016 is not just the absence of a middle ground, but the absence of political leaders who speak for it. Brexit has had no John Hume to lay the ground for a pragmatic compromise, and no Ian Paisley and Martin McGuinness to deliver it, either. Unlike in Northern Ireland’s, ours is a conflict which no leader has won the authority to try to resolve by consensus. And we shall pay the price for that for years.

• Martin Kettle is a Guardian columnist