Labour has urged the government to either delay Brexit, embrace a customs union or let MPs decide a way forward, during a prime minister’s questions fronted by stand-ins for Theresa May and Jeremy Corbyn.

With May holding talks in Northern Ireland, David Lidington, the Cabinet Office minister and de facto deputy prime minister, appeared in her place. Standing in for Corbyn was Emily Thornberry, the shadow foreign secretary.

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Thornberry used all her questions to probe Lidington on what she said should be a “sensible, grownup discussion” about the next steps forward on Brexit. Their exchanges were notably more polite than the usual barbs exchanged by the prime minister and Labour leader.

Thornberry was nonetheless scathing about May’s hopes of securing sufficient changes to the Irish backstop to assure her backbenchers during talks in Brussels later this week, saying she was running out of time.

“I hope he will understand the concern all of us have, not just in this house, but across the country, that we have a government treading water in the Niagara River while the current is taking us over the falls,” she said.

Facebook Twitter Pinterest David Lidington stands in for Theresa May at PMQs. Photograph: House of Commons/PA

Thornberry began with a joke about Lidington’s role: “As the newspapers always call him, he is ‘effectively the deputy prime minister’, surely the only occasion these days when the words ‘prime minister’ and ‘effective’ are used in the same sentence.”

After questioning Lidington about when MPs would next get a meaningful vote on any Brexit plan, Thornberry said it appeared this would happen only if and when May secured concessions on the backstop from the EU.

She said: “Does the prime minister seriously think that she will get anything different than the responses we have heard from the EU over recent days? Because none of them have given any encouragement that they are willing to reopen the withdrawal agreement unless the prime minister is willing to reconsider the red lines on which the agreement is based.”

The only “sensible, cautious thing to do at this late stage” was to seek an extension to article 50 to delay Brexit, she argued.

Lidington rejected the idea, saying it “would simply defer the need for this house, including the opposition frontbench, to face up to some difficult decisions”.

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That committed the UK and the EU to ensuring there was no divergence between Northern Ireland and the Republic of Ireland.

But it also, after an intervention by the Democratic Unionist party, committed the UK (not the EU) not to have any trading differences between Northern Ireland and Great Britain.

The problem is that these are two irreconcilable agreements. They also impinge on the legally binding Good Friday agreement, which brought peace to Northern Ireland and in some senses pooled sovereignty of Northern Ireland giving people a birthright to be Irish or British or both.

If the UK leaves the EU along with the customs union and the single market then the border in Ireland becomes the only land border between the UK and the EU forcing customs, tax and regulatory controls. The backstop is one of three options agreed by the EU and the UK in December and would only come into play if option A (overall agreement) or option B (a tailor-made solution) cannot be agreed by the end of transition. The Irish have likened it to an insurance policy. The new EU idea is to extend the transition period to allow time to get to option A or B.

But an extension is problematic for Brexiters and leave voters, who want the UK to get out of the EU as soon as possible.

The Irish and the EU will also still need the backstop in the withdrawal agreement, which must be signed before the business of the trade deal can get under way. Otherwise it is a no-deal Brexit.

Extending the transition into 2021 would mean another year of paying into the EU budget. Britain would have to negotiate this but it has been estimated at anywhere between £10bn and £17bn. Staying in the EU for another year would also mean continued freedom of movement and being under the European court of justice, which Brexiters would oppose.

Ideas such as using technology to prevent border checks would never work, argued Thornberry. She said: “The technological solution is a non-starter; a permanent backstop will never be acceptable to the ERG [European Research Group] or the DUP; and the only solution that will actually work is a full customs union.

“It is the answer that is staring the government in the face. If they backed it, it would command a majority in this house. It would avoid the mayhem and the chaos of no deal, and it would protect the jobs at Nissan, Airbus and elsewhere which are currently at grave risk. So can the minister explain why the minister is so dead against it?”

A customs union on its own would not be enough, Lidington responded, saying it would not cover areas such as common regulatory standards for goods or the movement of food and livestock.

Thornberry praised Lidington as the minister who understood Europe “better than any of his cabinet colleagues”, but said not even he had been able to properly explain the government’s plans.

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“The truth is there are no answers,” she said. “Plan A has been resoundingly rejected by parliament. Plan B was ruled out by the EU months ago, and the government is in danger of sleepwalking the country towards leaving with no plan and no deal at all.

“So with just over 50 days to go, can I give the minister a final opportunity to tell us whether there is a better plan than this, or for goodness sake will he let parliament take charge instead?”

Lidington said the answer lay in part with Labour. “If they are worried about no deal they have to vote for a deal, and every time they vote against a deal the risk of no deal becomes greater,” he said. “It really is time for the opposition frontbench, for once, to put the national interest first, do the right thing and vote for a deal.”