But after the prime minister personally assured a group of around 15 rebels that Mr Grieve's amendment would be partially adopted into the Bill before it went back to the Lords the rebels backed down.
But it was in the Labour Party where the deepest rifts were exposed.
"MPs are now voting to remove another exit day amendment".
Although the vote was technically a victory for May, her political sway appeared severely weakened as two competing wings of her Conservative party bickered over how close Britain ought to remain to the EU. Part A said that if parliament rejects the final Brexit deal, the government would have to set out a new approach within seven days.
326 voted against a customs union with EU, while 298 voted for it. Then her foreign minister, Boris Johnson, was recorded saying there could be a Brexit meltdown.
Other changes insisted upon by the Lords relating to the Charter of Fundamental Rights, principles of European Union law to be retained after Brexit and European Union environmental principles were also removed.
Carthy said, "In December the British government agreed that, in the absence of a comprehensive trade and customs agreement that avoids a hardening of the border in Ireland, a "backstop" will kick in".
On Tuesday, parliament will also debate other amendments, including a challenge to the government's plan to put March 29, 2019, or "Brexit Day", into law and an attempt to toughen a commitment to ensure a frictionless border between Northern Ireland and the neighbouring Irish Republic, which will remain in the EU.
While in the end, only two Tory MPs - Ken Clarke and Anna Soubry - voted against the government, there were clashes over how much of a say Parliament should get as the United Kingdom leaves the European Union, with one side accusing the other of trying to "wreck" Brexit - and being accused in turn of being "zealots" who wanted to sideline Parliament.
The senior Remainer said he hoped a compromise would be found and warned "this isn't the end of the matter" if that did not happen.
Labour leader Jeremy Corbyn.
Like the Conservatives however, the Labour Party has committed itself to restricting the free movement of people between Europe and Britain, violating one of the "four freedoms" which Brussels insists are non-negotiable for membership of its institutions. The votes were won quite comfortably in the end - or as comfortable as you can be with only 316 MPs, ten short of a majority. However, the party is divided on the issue, with a total of 89 Labour MPs choosing to defy Corbyn.
Labour opposed the EEA model and introduced its own amendment, tabled by Sir Keir Starmer, which was defeated by 82 votes on Wednesday night.
Laura Smith, the MP for Crewe and Nantwich, quit her job as shadow defence minister to vote against the EEA while Ged Killen, Tonia Antoniazzi, Anna McMorrin, Ellie Reeves and Rosie Duffield stepped down as parliamentary private secretaries to vote for it.
Tory MP and former education secretary Justine Greening, who supported Remain in the European Union referendum, told Sky News in response: "I think what we all need to recognise is that the positions people are taking on Brexit are genuine and passionately held - and should be respected".