May’s initial deal was crushed by a humiliating defeat in Parliament two weeks ago, but she survived a subsequent no-confidence challenge on a party-line vote.

The famously dogged — or obstinate — May appeared to be out of fresh ideas when she returned to the House of Commons last week with a “Plan B” that sounded a lot like a warmed-over Plan A.

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Parliament remains gripped by deadlock, without a consensus on how to exit the European Union after four decades of free trade and shared governing.

The flamboyant, sharp-tongued speaker of the House of Commons, John Bercow, has upended tradition by allowing a raft of possible amendments to be debated — leading his critics to charge that the speaker is trying to help backbench renegades foil Brexit by taking control away from the government.

One of the most popular amendments likely to be considered Tuesday, written by two backbenchers from opposing parties, seeks to give May until the end of February to secure a deal with Brussels that could pass the Parliament. If the prime minister fails again, then the chamber wants her to seek permission from E.U. leaders to delay Brexit beyond the scheduled departure date of March 29.

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There is disagreement over whether such a delay should be a few months long — or until the end of 2019, as the amendment proposes.

Another amendment seeks to stop May’s government from allowing Britain to crash out of the E.U. in two months’ time with no deal at all.

The no-deal scenario is unpopular among many members of Parliament, but it remains a real possibility. Governments and businesses in Britain and across Europe are preparing to spend billions on contingency plans.

Richard Harrington, an undersecretary for industry and energy, told a gathering in London that a no-deal Brexit would be “a total disaster for the economy.”

“I am very happy to be public about it and very happy if the prime minister decides I am not the right person to do the business industry job,” he said, essentially daring May to fire him.

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Yet some hardcore Brexiteers support leaving the European trading bloc with no deal, because of their extreme dislike of May’s approach, which seeks to keep Britain closely tied to European rules.

Many ordinary citizens who support Brexit are telling pollsters the same thing, that they are sick and tired of the endless dithering and just want out.

Andrea Leadsom, the Tory leader of the House, told the Sunday Times that “taking no deal off the table has been used as a thinly veiled attempt to stop Brexit.”

Recent days have seen British business leaders warn that a no-deal scenario poses real risks to the economy.

The British Retail Consortium, which includes major grocery store chains, warned Parliament that it is reliant on fresh produce from growers in the European Union — and that it would be impossible to stockpile greens and fruits if Britain crashes out of the trading bloc with no deal.

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“We are extremely concerned that our customers will be among the first to experience the realities of a no deal Brexit,” the association said in a letter reported by the BBC.

Airbus chief executive Tom Enders said last week, “Please don’t listen to the Brexiteers’ madness, which asserts that ‘because we have huge plants here we will not move and we will always be here.’ They are wrong.”

Airbus employs 14,000 people in Britain, manufacturing airplane wings.

“It is a disgrace that, more than two years after the result of the 2016 referendum, businesses are still unable to plan properly for the future,” Enders said.

His remarks came as the company that assembles Jaguars and Land Rovers, Britain’s biggest carmaker, said it would extend its annual spring assembly-line stoppage for an extra week because of Brexit uncertainties.

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Facing such dour warnings from British business, May will probably try to persuade Parliament to allow her to return to Brussels one more time to try to strike a better deal.

In a briefing with reporters on Monday at 10 Downing Street, May’s official spokesman, who goes unnamed according to protocol, said that the prime minister wants two more weeks to try to change the withdrawal agreement she spent two years negotiating in Brussels. Another vote on her new and improved Brexit deal, if she can strike one, would be held Feb. 14.

The most contentious section of the failed agreement involves the legally binding guarantee, the “backstop,” which is designed to prevent a hard border between Northern Ireland and Ireland — essentially between the United Kingdom and the E.U. — after Brexit.

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The Irish border backstop requires Britain to remain closely aligned with E.U. rules and customs arrangements if it is unable in the future to agree a new free-trade deal with the bloc.

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Critics say the provision could trap Britain in the E.U. forever.

Boris Johnson, a leading Brexiteer and former foreign secretary who has previously argued that Britain should have the courage to leave with no deal, now appears to be tacking.

If May were able to secure a “freedom clause” from Brussels that would make the backstop time-limited or allow Britain to leave on its own, without permission from the E.U., she would win the “full-throated” approval of the entire nation, Johnson wrote in his weekly column in the Daily Telegraph.

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This won’t be easy.

As reporters were being briefed at 10 Downing, Margaritis Schinas, the European Commission spokesman, told journalists in Brussels that the withdrawal agreement now on the table has the unanimous backing of the leaders of the 27 nations remaining in the European Union.

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“It is endorsed by leaders and is not open for renegotiation,” Schinas said.