The leading Tory Eurosceptic highlighted areas of the deal where he said the “honourable” Prime Minister had reneged on promises over leaving the customs union, maintaining the internal integrity of the UK and leaving the jurisdiction of the European Court of Justice.
Speaking in the Commons, chairman of the European Research Group Mr Rees-Mogg added: “As what my right honourable friend says and what my right honourable friend does no longer match, should I not write to my right honourable friend the member for Altrincham and Sale West?”
This was a reference to Sir Graham Brady MP, the chairman of the Tory 1922 committee, to whom MPs must write to express no confidence in a leader in order to trigger a challenge.
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Mrs May replied that “some difficult choices have had to be made” to avoid a hard border on Ireland, adding: “It is not only our intention, but we will be working to ensure that protocol does not have to be put into place.”
Tory divisions further burst into the open when former minister Mark Francois urged Mrs May to “accept the political reality of the situation” as he noted more than 80 Conservative colleagues opposed her deal.
Sir Nicholas Soames, a Tory former minister, could be heard saying “sit down” as Mr Francois told the Commons: “Prime Minister, the whole House accepts that you have done your best.
“But the Labour Party has made plain today that they will vote against this deal, the SNP will vote against it, the Liberals will vote against it, the DUP will vote against it - our key ally in this place will vote against it, over 80 Tory backbenchers - well, it’s 84 now and it’s going up by the hour - will vote against it.
“It is therefore mathematically impossible to get this deal through the House of Commons.”
Mr Francois added the deal was “dead on arrival” before the PM made her speech, adding: “I plead with you to accept the political reality of the situation you now face.”
Tory Remainer Anna Soubry earlier asked Mrs May to put a second referendum back on the table because the promises made during the first referendum “can’t be delivered upon”.
She said: “We face three choices. We either accept this agreement - and I respectfully suggest there is now no majority in this House for it - or we accept no deal - which would be profoundly irresponsible and catastrophic for this country.
“Or we have no Brexit. We remain in the EU, the best deal that we have with the EU, and on that basis would she undertake not to rule out taking this back to the British people and having a People’s Vote?”
Mrs May rejected the notion, saying she was not going to change her mind.
Conservative former education secretary and Remain-supporter Justine Greening earlier said she did not believe the draft deal was good for Britain’s “long-term future”.
She told the Commons: “The Prime Minister said this is in the national interest, so why not allow people in our nation to have their say now? If it was good enough before, why isn’t it good enough now?”
Labour former minister Chris Leslie, a prominent Remainer, said: “It is quite clear, we’ve been going for about an hour now, not a single honourable or right honourable member has supported the plans that the Prime Minister has set out. So it’s quite clear she cannot command the House of Commons on these proposals.”
He said he was tempted to ask Tory MPs to put their hands up if they support the PM, but no hands were raised.
Tory Brexiteer Sir William Cash (Stone) said the 585 pages of the draft deal were a “testament to broken promises, failed negotiations and abject capitulation to the EU”.