MPs voted to pry Britain’s faltering Brexit negotiations out of Theresa May’s rigid grip last night, backing an amendment that will allow them to hold a series of “indicative votes” to try to find a solution to the UK’s constitutional crisis.

The humiliating defeat led to renewed calls for the Prime Minister to resign.

Last week the EU allowed Britain to delay Brexit until May 22, but only on the condition that MPs voted for the Tory leader’s deal.

If that doesn’t happen – and it’s still not entirely clear if the Prime Minister will put her agreement to a vote again – then the UK only has until April 12 to come up with an alternative solution.

Tory grandee Sir Oliver Letwin, who was behind the amendment, said his proposal would help find that alternative. It would, he said, be about solutions rather than “unicorns”.

The Government had hoped to defeat Letwin’s motion by promising a day of debate to consider what would happen if no Brexit deal can be agreed.

But there is little trust in the Commons for the current administration.

Scotland’s First Minister Nicola Sturgeon had urged MPs to back Letwin’s amendment. She tweeted: “Last week, the EU saw that the PM had no coherent or deliverable #Brexit plan and decided to come up with one for her. The Commons must do the same tonight.”

Business Minister Richard Harrington resigned his position in Government to vote for the amendment.

In a brutal resignation letter, he accused the Government of “playing roulette with the lives and livelihoods of the vast majority of people in this country who are employed by or otherwise depend on businesses for their livelihood”.

He said firms had told him that the Government’s chaotic Brexit had led to “cancelled investment decisions, business being placed abroad and a sense of ridicule for British business, across the world”.

Foreign Office Minister Alistair Burt and Health Minister Steve Brine also backed the amendment, and resigned from the Government.

In his speech, Letwin put forward a plan for how Wednesday should work, He suggested a first vote to enable MPs to disclose “where the votes lie, on a plain vanilla basis” – with all votes cast on a slip in the voting lobbies at the end of the debate – before seeking to “zero in on something which could be a compromise” that could secure a majority in the following days.

He added: “It is also the case we will have to attend to the question ... which is what will the Government do if the House reaches a majority view, not for some unicorn, not for some ludicrous proposition which utterly contradicts common sense, but for a sensible way forward.

“How do we persuade at that stage the Government to allow that majority view to be implemented – that will be a major issue.”

Shadow Brexit secretary Sir Keir Starmer said the Prime Minister had got herself and the Government into a “hopeless position”.

He said: “Having disregarded views from across this House for the best part of two years, the Government now finds itself with a deal it just can’t get through this House and time has almost run out.”

May, he said, had lost control of the negotiations and appeared to have lost control of her party.

Speaking after the vote, the SNP’s Tommy Sheppard said it was time for the Prime Minister to go: “This is major blow to government. A huge indictment upon and censure of incompetent Tories. May should do the decent thing and go.”

The chaos in Westminster came as the European Union announced that it had now completed its preparations for a no-deal Brexit.

In an ominous statement, the European Commission said: “As it is increasingly likely that the United Kingdom will leave the European Union without a deal on 12 April, the European Commission has today completed its ‘no-deal’ preparations.”

“While a ‘no-deal’ scenario is not desirable, the EU is prepared for it,” the commission’s statement added.