“Even five years ago, if I had thought I would be sitting in my room every few days with the son of Neil Kinnock, I would have said you were living in a parallel universe. And now, I’m talking to him all the time.”

The Conservative MP Robert Halfon almost seems surprised at himself as he sits chatting in his Westminster office, the walls lined with political portraits, cartoons and newspaper clippings.

Brexit has shattered lifelong friendships and reopened old fractures across both main political parties. But as MPs have grappled with the agonies of the past two years, new and unlikely alliances have sprung up, too.

Halfon and Stephen Kinnock, the son of the former Labour leader, are one half of an incongruous quartet – the other two being Labour’s Lucy Powell and the Conservative Nick Boles – who have spent months working together to try to forge a majority in the House of Commons for a deal they call “common market 2.0”.

They hold regular meetings here to plot their next move while maintaining close links to wider groups of supporters in their respective parties and with other key players in the parliamentary drama around Brexit, such as Yvette Cooper and Hilary Benn.

Now, after Theresa May’s deal suffered a second calamitous defeat this week, these four new-found friends hope the space may finally open up for a cross-party consensus to be found.

They chatter constantly as the Guardian’s photographer arranges them; at one point, Powell dashes off to the chamber of the House of Commons.

It’s Thursday morning, and they are discussing how to maximise support for Benn’s amendment, to allow parliament to take control of the next steps of the Brexit process.

Later that day, the amendment narrowly falls, but only after they have extracted a pledge from the government that, if May’s deal is rejected for a third time next week, parliament will be allowed to vote on alternatives on 25 March.

Boles says: “I think the week of the 25th is our high noon, even if the PM’s high noon has already passed.”

The four hope some of those MPs whose first choice would be another Brexit referendum, or who were willing to support May’s deal – as Boles and Halfon did last week – may now be ready to consider what their second-best option might be.

Powell says: “To get anything through here, we need a good chunk of Conservatives, and we need most of all the opposition benches. It’s just very careful building of alliances, and having conversations - and that’s a tough job, in this climate, and that’s what we’re trying to do.”

Kinnock was an early convert to the common market proposal – also known as “Norway-plus” – under which the UK would join the European Free Trade Area (EFTA) and, in that way. regain membership of the EU single market.

Quick Guide What is the common market 2.0/Norway-plus Brexit option? Show This soft Brexit compromise has been championed as a plan B for leaving the European Union. It is based on Norway’s relationship with the EU, which is outside the bloc and the customs union but inside the single market. Under the plan the UK would have to join Norway, Liechtenstein and Iceland in the European Free Trade Association (Efta), which would then allow it to participate in the European Economic Area (EEA). The ‘plus’ in this option refers to a temporary customs union with the EU, which would need to be negotiated to avoid a hard border on the island of Ireland. This arrangement would remain in place until the EU and UK agreed a specific trade deal. The option has the advantage of being as close to the EU as possible without full membership, and it would do away with the need for the problematic backstop for Northern Ireland. Like Norway, the UK would be outside the common fisheries and agriculture policies, and would not be subject to the European court of justice. But it crosses a key red line for Brexiters by continuing freedom of movement, one of the preconditions of single market membership. It would also limit the UK's ability to negotiate its own trade deals while a new customs arrangement is under discussion. And it would require continued financial contributions to the EU without an influence, as the UK would no longer have MEPs or a seat on the European Council. It also isn't entirely clear that the UK would be welcomed into Efta.

The government would then seek to negotiate a customs arrangement to avoid border checks in Northern Ireland (most of which, they point out, would be obviated by single market membership anyway).

Kinnock says: “I lived and worked for many years in Brussels and I know how the EU operates, and felt right from the beginning that they would go for a model, and not some cooked-up thing. That was the first thing I thought: it’s got to be off the shelf.”

Boles, too, was an early advocate; Halfon calls him “the John the Baptist in our party on all this”.

Halfon himself came to the idea more recently: “I started reading all these articles about EFTA and the EEA in the runup to the end of last year, October, November. It was literally a eureka moment. I thought: why haven’t we done this?”

They admit that they have been indebted to Powell, the Manchester Central MP who was Ed Miliband’s campaign chief, for helping them to communicate the idea, particularly to Labour MPs.

Kinnock, Powell and Boles, together with the former Tory minister Sir Oliver Letwin, who has also been liaising with Downing Street, held an unlikely meeting with Jeremy Corbyn earlier this month.

“We had detailed, forensic discussions. They all had the pamphlet in front of them, heavily annotated. Very good questions, very good discussion,” Kinnock says. “They are, I hope, minded to give us the whip.

“What we then need is the right number of Conservative MPs, alongside the SNP [Scottish National party] and others to get us over the line.”

All four are united in their concerns about the risks of a second Brexit referendum.

Kinnock says: “I think the country is deeply, deeply divided. To argue that you can reunite the country by having another referendum, that argument just doesn’t stack up for me. To me, that’s like saying, my house is on fire, and I’m going to put the fire out by pouring petrol on it.”

Halfon says he would “die in a ditch” to prevent what he calls the “political calamity” of a referendum.

“We would have a host of Tommy Robinsons in parliament, I think, if we had a second referendum – it would be a disaster for trust in our political system,” he says.

Powell says her scepticism about a referendum – a cause enthusiastically embraced by many Labour MPs who are her natural political allies – has sometimes made life difficult. “It does put me in a different position to a lot of my friends and that can bring new tensions.”

None of the group believes they will ever follow the example of Chuka Umunna and Anna Soubry, whose anti-Brexit alliance became so strong that it led to both leaving their respective parties to form the breakaway Independent Group.

But they do say their collaboration has paved the way for future cross-party wor – which will be a headache for party leaders for years to come.

Powell says: “I definitely think traditional whipping has taken a major knock and it’s hard to see that snapping back into place, because you’ve got people who have just become accustomed to making up their own minds.

Kinnock says: “I would disagree on many many issues with Rob, and with Nick. But when it comes to a moment where the nation is in a political crisis, and potentially on the brink of a culture war, we as politicians have to step up and we have to show that those narrow, tribal differences, need to be left at the door.”

“It’s been a real pleasure,” says Boles. “We don’t pretend not to be different. We don’t pretend not to have different philosophies, different priorities: and if anything we actually take some pleasure in finding out more about the others. It’s been a great experience, almost whatever the result.”