Here we go again: Britain’s Parliament is debating Brexit and Prime Minister Theresa May is playing for more time as various factions try to influence the endgame. Her goal is to deliver her divorce deal without dividing her party. So what’s new?

For one thing, the time left on the clock. Just 45 days remain until Britain exits on March 29 — unless both sides agree to an extension. If nothing happens, the UK will leave without a deal. The potential for such a major political and economic shock concentrates minds.

The bigger change is that Britain’s Labour Party has finally joined the Brexit debate for real. For the past two years, it has been happy to sit back and throw popcorn at the Government while the “Nightmare on Downing Street” unfolds. Brexit was a Conservative project and Labour, just as divided on the issue, wanted the Government to own the mess.

Jeremy Corbyn has finally woken up to the reality that his party is in the same bad movie as the Government. His new positions aren’t any less cynical — but no matter. They change the calculus facing the Government, the Conservative Party and even the European Union in the crucial weeks ahead. Ironically, they make it more likely that May will get her deal, if you believe the outcome must be rational; they raise the risk that it will all go wrong if you believe that emotions and party loyalty will prevail.

Labour shifted its policy in two key areas last week. First, it has dropped its insistence that any Brexit deal should deliver the “exact same benefits” as actual European Union membership. That, of course, was never possible.

It has also quietly abandoned its opposition to the Irish backstop, the controversial provision in May’s deal that could tie Britain into the EU customs union indefinitely.

Last year, Corbyn was clear that “there certainly wouldn’t be a backstop from which you can’t escape”. But, in a letter to the Prime Minister last week setting out his five demands for Brexit, he acknowledged that “any withdrawal agreement would need to include a backstop”.

Corbyn’s main requirement is that the UK commits to stay in the EU’s customs union and continues to be bound by the bloc’s rules in a number of areas. It is a call for the softest of Brexits consistent with leaving the EU’s single market and ending free movement. Remember that many Labour “leave” voters were persuaded by anti-immigrant arguments in the referendum.

Set aside for a moment that all of Labour’s demands are perfectly possible within the existing wording of the non-binding political declaration that accompanies May’s deal. There is nothing to stop Britain from joining the EU’s customs union or aligning with the single market’s rules. Ignore too that Labour want the UK to have a say in the EU’s trade agreements with other countries; if that happens, it will be mere tokenism.

For Corbyn, it’s a sweet spot. A euroskeptic since the 1970s, he has no desire to cancel Brexit. His party manifesto includes a commitment to delivering on the 2016 vote. Many remainers in his party would like a second referendum, which is also official Labour policy, if there isn’t a General Election. But, as British commentator Robert Peston has put it, the socialist leader “is less attracted to a referendum than he would be to a job offer from Goldman Sachs”. Never quite the democrat he pretends to be, Corbyn is quietly burying the idea of a second referendum.

What’s important is that with Corbyn’s offer, a parliamentary majority exists for a Brexit deal that reclaims control over Britain’s borders, a key requirement for May and many leave voters, and which removes the need for a backstop. That also lessens any pressure on the EU to come up with a game-changing concession.

Conservative Brexiters, however, are fixated on the idea of Britain negotiating its own trade agreements after Brexit; being in the customs union would make that impossible. The problem for May is that a cross-party agreement like that would lose the support of Brexiters in her own party — and put the party at risk of splitting. Cross-party coalitions are still a rarity in British politics, with the exception of when the issue is the country’s relationship with Europe.

Unsurprisingly, May has so far rebuffed Corbyn. But Labour and government officials are still talking and she is promising handouts and concessions on areas like workers’ rights to woo the Opposition.

The suspicion among many is that May is simply running down the clock. Her gamble is that, by the end of this month, the two alternatives to her deal — no deal or no Brexit — will prove unacceptable and she’ll get her way. Tomorrow’s vote in Parliament on amendments rather than the Government’s deal — is unlikely to box her in.

She may be right, but the wait for businesses and residents will be terrifying. Figures published this week showed how Brexit uncertainty is taking a toll on the UK economy.

However the negotiations and votes unfold, the Government and Opposition will need to be aligned on two overriding goals: avoiding a chaotic, no-deal exit and preserving the letter and spirit of the Good Friday Agreement that brought peace to Northern Ireland. At the moment, the only ways to satisfy both while delivering on Brexit — which both leaders say they want — are Corbyn’s proposal or May’s deal, spiced with whatever the EU feels it can give her.

If May’s Conservatives reject both, the Government must be willing to contemplate putting the question to the public, either in a referendum or a new election.

• Therese Raphael writes editorials on European politics and economics for Bloomberg Opinion. She was editorial page editor of The Wall Street Journal Europe