“We are going at a snail’s pace, but at least the snail is moving,” was how one Labour frontbencher described the tortuously slow evolution of his party’s policy on Brexit last week. “The question now is, can we hurry the bloody thing along to where it needs to be before it is too late?”

After months of taking refuge in what has become known in Labour circles as “constructive ambiguity” – deliberately blurred positions that were designed to keep both Remainers and Leavers onside – party leader Jeremy Corbyn appears to be shifting (albeit relucantly) towards a more soft-Brexit position.

Nothing about the process has been easy or quick – not even settling on the location of a so-called shadow cabinet “awayday” to thrash out the latest policy change. After being delayed several times, and the venue switched, the awayday eventually took place on home territory last Monday in Corbyn’s office in the House of Commons.

On Monday the Labour leader will announce the outcome of those deliberations in a keynote speech at the National Transport Design Centre at Coventry University. Corbyn is expected to confirm that Labour wants to keep the UK in some form of customs union permanently, after the post-Brexit transition period ends in 2021.

“Jeremy and his people will try to play this down,” says one shadow cabinet source. “He is a lifelong Eurosceptic. But it is very significant. It moves us decisively away from the Tories and away from hard Brexit. It rejects the Tory model of global Britain, under which they would spend years trying to agree trade deals across the globe. And it will sort the Northern Ireland problem out, which otherwise would be a nightmare.”

Labour’s latest move – urged upon Corbyn by a majority of his shadow cabinet and an increasingly rebellious group of backbenchers – is already causing political tremors across Westminster, and spreading panic in Tory ranks.

With the official opposition now in favour of a customs union, and at least a dozen Conservative rebels led by Anna Soubry suggesting they will put their names to, and vote for amendments to reach that end, there is now very likely to be a Commons majority in favour.

The power Labour MPs have, to inflict potentially devastating defeats on Theresa May by delivering cross-party parliamentary votes for a soft Brexit with Tory Europhiles, is becoming clearer by the day.

Chequers away-day: May met cabinet ministers to talk about the EU future partnership. Photograph: Jay Allen/MoD, Crown Copyright

The former Labour minister and anti-Brexit MP Chris Leslie believes the political dynamics of the debate at Westminster are changing as parliament takes control and party loyalties give way to the national interest. “Hilary Benn [Labour chair of the all-party Brexit select committee] summed this up when he talked about this being a backbenchers’ parliament. He is right. There is a dawning realisation about what we can achieve for this country if clusters of MPs across parties abandon tribalism and work together on smart ideas. Unquestionably, there is a parliamentary majority for a soft Brexit and as we realise what we can achieve, it whets the appetite to do more.”

That explains why today more than 80 senior Labour figures including 37 MPs, 12 MEPs, a string of Labour leaders in local government, trades unionists and peers (the former party leader Neil Kinnock included) have issued a statement to the Observer demanding that Corbyn does not stop at staying in a customs union, but goes a big step further and comes out in support of the UK remaining as a full participant in the EU single market.

To make such a demand on the eve of Corbyn’s big speech will be seen by some as highly provocative. But the group will answer that charge by saying the issues at stake are too important and too urgent. They also say they are trying to help Corbyn, by arguing that he will not be able to deliver his anti-austerity agenda, and better-funded public services if the UK has to take the economic hit that will inevitably result from leaving the single market. They also explicitly shoot down the traditional leftwing view of the single market as a “capitalist club” and instead portray membership as essential to the causes of social justice and workers’ rights – both near the top of Corbyn’s agenda.

Sir Keir Starmer, the shadow Brexit secretary, has not ruled out Britain staying in the single market. Photograph: Graeme Robertson/The Observer

Labour’s Brexit spokesman, Keir Starmer, has not ruled out remaining in the single market, but says doing so must be conditional on the EU agreeing to change rules on free movement of workers so that the UK can have greater control over immigration.

A senior Labour MP who is pushing the pro-single market cause says that, were Corbyn to move again, there would be a clear majority in parliament for staying in the single market too. “We have a long way to go to persuade the leader on this. But we have to push it. Parliament would back it if there were free votes.

“There are many Tories who would prefer to stay in the single market. The vast majority of Labour MPs would, the SNP does, the Liberals do. If we could shift Corbyn on this, we could deliver in both houses of parliament.”

On Thursday, media attention was focused on Conservative divisions on Brexit, as Theresa May summoned 11 cabinet ministers for an awayday in the wood–panelled spendour of Chequers. Hardliners Boris Johnson and Michael Gove were supposed to agree with soft Brexiters like Philip Hammond and Amber Rudd what the UK’s post-Brexit relationship with the EU should look like. It was always going to be an impossible task.

Anti-EU Conservative backbenchers, now led by Jacob Rees-Mogg, prepared to pounce on any sign of compromise, any hint of a betrayal of their vision of pure Brexit. As a result, little sign of meaningful movement emerged, other than suggestions that the cabinet wants to maintain “frictionless trade” in some sectors by aligning with EU regulation, while diverging in others to gain a competitive advantage in international markets.

Theresa May is due to make a speech on Friday, in which she is expected to flesh out this approach, but it was pre-emptively dismissed by Brussels as another repackaged UK demand to cherry-pick the best fruits from the EU orchard after Brexit – something Brussels will not allow. “It is not possible for the UK to be aligned to the EU when it suits and not when it doesn’t,” said Leo Varadkar, the Irish taoiseach.

May, lacking a majority in parliament, is trapped between rival factions in her own party who will not allow her to move one way or the other without calling her leadership into question. Since the prolonged, bitter rows over the Maastricht treaty in the early 1990s, it has been the Eurosceptic Tory right that has caused most problems for Conservative prime ministers over the UK’s relationship with Europe. But as Labour begins to shift and large numbers of anti–hard Brexit MPs show they are prepared to abandon tribalism, new alliances are forming in parliament. If enough soft Brexit Tories are prepared to join Labour, SNP and other colleagues, it could on this occasion be the Tory left that becomes the decisive force in the backbenchers’ parliament, tipping the scales in the crucial votes to come. If it does, expect a very different Brexit to the one desired by the likes of Johnson and Rees-Mogg.