It means the Brexit strategy of both Theresa May and Jeremy Corbyn has been blown apart in the past 24 hours.

The rebel Labour amendment in the Lords opened the prospect of a Commons vote on the EEA – a less stringent version of the single market – after 83 Labour peers voted against the party whip.

Chuka Umunna, the Labour MP who co-chairs a pro-European Commons committee with the Conservative Anna Soubry, said the leadership would now have to come off the fence and make it clear where it stood.

“The time for constructive ambiguity is over – our members and our voters will be delighted with this clear signal that we will not go along with this Tory Brexit,” he said.

The vote came hours after Boris Johnson called the prime minister’s proposal for a customs partnership “crazy” in an interview with the Daily Mail that dealt a major blow to the government’s strategy of a cautious balance between leave and remain.

Johnson was the subject of fierce criticism from within Tory ranks all day. In the Lords debate the former Tory MP Patrick Cormack asked: “What sort of example are we being given by a cabinet that is rent asunder by the foreign secretary, the second most important cabinet minister, rubbishing the prime minister in the columns of the Daily Mail?”

On the key EEA amendment, Labour peers were whipped to abstain. But 83 defied the whip to back the amendment, including a former party leader, many former ministers and a former chief whip. Seventeen Conservatives also backed the amendment.

It was the third defeat of the afternoon for the government and an unexpected triumph for a cross-party group that included Waheed Alli for Labour, the Conservative peer Sandip Verma and the crossbench peer Karan Bilimoria.

All the amendments to the bill that have been passed in the Lords will have to be considered and voted on by MPs when the bill returns to the Commons, perhaps as soon as next week. 

Lord Alli told the House that continued membership of the EEA was vital to ensure the future profitability of the UK’s export business and the jobs and livelihoods of many thousands of people.

“It is the EEA that deals with services, services like retail, tourism, transport, communications, financial services and aerospace where we have a £14bn trade surplus,” he said.

“The customs union only will benefit our European neighbours in their imports and without an EEA equivalent it will damage our profitable export business.”

In a rowdy and sometimes ill-tempered debate peers, many of whom had been debating the EU for the past 30 years, argued passionately for and against EEA membership.

Officially Labour opposed the Alli amendment because other amendments gave MPs a role in the negotiations.

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But the leadership will now have to face the tension between its strongly pro-remain rank and file membership, its largely pro-remain MPs and the policy consequences of remaining in any arrangement with Europe such as the EEA, which limits its plans for greater state involvement in the economy.

The Tories are now likely to try to brand Labour as the party of free movement, which is one of the obligations of EEA membership. The threat of higher EU migration is likely to alarm Labour voters in some parts of the country.

A Department for Exiting the European Union spokesperson said: “The referendum was a vote to take control of our borders, laws and money. Ongoing participation in the EEA would mean having to implement new EU legislation automatically and in its entirety without having a say on how it is formulated – and it would also mean continued free movement. We will now consider the implications of this decision.”

Peter Mandelson, the former trade secretary and one-time EU trade commissioner, said ministers were perpetrating a “Brexit fraud’ by pretending that migration from the EU, which would continue under membership of the EEA, would end after the UK left.

“The time has come for economic reality and commonsense to prevail over wishful thinking and political dogma,” he said. “This gives us the opportunity to do the right thing for the country and in my view that is what we have a duty to do.”

The EEA offers most of the benefits of the single market, without being subject to the European court of justice. It does not cover the common agriculture or fisheries policies.

Earlier, peers voted to remove the EU exit date of 29 March 2019 from the withdrawal bill, warning that it would be a straitjacket for negotiators. The Duke of Wellington, Charles Wellesley, who moved the amendment, insisted he was not trying to undermine the result of the referendum by delaying or averting Brexit.

“We should give ministers a bit more flexibility to secure and obtain ratification of the best possible deal, which will do the least damage to the economy and the national interest,” he said.

The government also lost another vote on an amendment seeking to protect UK membership of EU agencies such as Euratom. 

The Labour leader in the Lords, Angela Smith, said it was an opportunity for the Commons to think again.

“The House of Lords amendment is not about stopping Brexit but the fine print of when and how the agreements are concluded,” she said.

The defeats came on the final day of debate on the report stage of the bill in the Lords. Peers are due to send it back to the Commons at the end of the week, and the government hopes it will be on the statute book before the end of May. 

The government has been forced to make concessions, as well as losing a dozen votes during the debates, in a series of amendments that have often had cross-party support.

The amendments have been divided between trying to preserve benefits of EU membership, such as workers rights, and other fundamental rights through the incorporation of the EU’s charter of fundamental rights, which has a broader reach than the UK Human Rights Act.

Others were intended to promote the place of parliament at the heart of the process of leaving the EU, including clarifying the significance and consequences of the “meaningful vote” on any Brexit deal.

And some were more narrowly technical, intended to restrict ministerial powers to change laws without proper parliamentary process and debate.

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