That backstop plan was an "insurance policy", meant to confirm that there will be no return to a visible border between Northern Ireland and the Irish Republic after Britain leaves the EU.
Many British lawmakers fear the backstop will trap Britain in regulatory lockstep with the European Union, and Parliament earlier rejected the Brexit withdrawal agreement May sealed in November with the EU.
But there was a warning from the independent Unionist MP for North Down, Lady Hermon, who called the Prime Minister's backstop plans "nebulous" and said she had a "duty to spell out" what the alternative arrangements are.
But May's victory on Tuesday may be short lived.
European Commission president Jean-Claude Juncker has rejected calls by MPs to revisit the Brexit Withdrawal Agreement, stating that it remains "the best and only deal possible".
Ireland also has a huge interest in preserving a deal that has maintained sectarian peace in neighboring Northern Ireland for two decades, not least by scrapping all border checks.
An attempt by Labour lawmaker Yvette Cooper and Conservative Nick Boles to give parliament the power to request a delay to Britain's March 29 exit was defeated by lawmakers on Tuesday, but Boles said he would renew that effort on February 14 if a deal has not been passed by then.
But May says she needs more - a legally binding change.
It is time to draw the first phase of the UK's withdrawal from the European Union to a close, and in order to do that we have to agree a deal with the European Union that gets us on to negotiating our future trading relationship.
The EU parliament's point-man on Brexit, Guy Verhofstadt, underlined that nobody in Europe wanted to use the backstop, but that it's "needed to be 100 percent sure that there is no border between Northern Ireland and the Irish Republic".
"What we are being asked to do here is to compromise on a solution that works, and to replace it with wishful thinking", he said.
While the European Union has repeatedly refused to reopen the divorce deal, European Union sources said additional clarifications, statements or assurances on the backstop might be possible.
"My message to PM @theresa_may: The EU position is clear and consistent".
A fourth rejected amendment - Amendment G - from former Conservative minister Dominic Grieve, would have allowed MPs to express their will in a series of votes right up to Brexit day.
It courts Brexiteers with a promise to ditch an unpopular Irish border policy in favour of an alternative, and appeals to remain-supporting Conservatives by pledging safeguards against the risk of disruption if no deal can be agreed with the EU.
After pleading for a mandate from Parliament to confront Brussels over the controversial Irish border "backstop", the United Kingdom government won over Conservative Brexiteers and secured a crucial Commons victory by 317 votes to 301.
On Wednesday, she met British opposition leader Jeremy Corbyn for what her spokesman said were "serious and engaged" talks on the way ahead. This could be an oppportunity for an eleventh-hour deal, or it would be the last chance to agree an extension of the Article 50 negotiation period and delay Brexit to avoid no-deal disruptions.