Trying to sort out Britain’s EU exit was the downfall of Mrs May, now Mr Johnson is battling to make sure he does not meet the same fate.
Having initially been bullish about scrapping the backstop, there has been speculation that the prime minister may be softening that position somewhat.
Sky News takes a look at the different versions of the backstop that have been proposed – and why the concept itself has proved so contentious.
First things first, what is the backstop?
The 2016 vote for Brexit has raised a multitude of questions, not least about the future status of the border between Northern Ireland – part of the United Kingdom – and the Republic – an EU member.
Because of this new disparity between two countries that share a land border, arrangements for avoiding a return to a hard frontier (border posts and checkpoints) need to be agreed.
The backstop is an insurance policy contained in the withdrawal agreement negotiated between Mrs May and the EU to avoid a hard border.
The original plan was for the withdrawal agreement to be passed by parliament and for Britain to enter into a transition period.
This is essentially a standstill period, in which Britain would continue to follow EU rules and regulations, with the intention being that this time would be used by government and businesses to get ready for Brexit to take full effect.
In this period both sides would negotiate the terms of the future relationship between Britain and the EU.
The hope was that, all being well, these talks would have worked out a way of ensuring there is no need for a hard border between Northern Ireland and the Republic.
If not, the backstop would kick in.
But things have not gone according to plan. Mrs May failed to get her deal through parliament and Brexit has been delayed twice.
Mr Johnson is now trying to find a way through the deadlock – and tweaking the backstop rather than getting rid of it altogether has been suggested as a potential compromise.
This is the current version of the backstop that is in the withdrawal agreement.
This would see the UK as a whole remain in a customs union with the EU, while Northern Ireland would follow further EU rules and regulations in order to keep the border frictionless.
In a development that will leave seasoned Brexit observers feeling like they’re in Groundhog Day, this idea has cropped up once more.
If it sounds familiar, that’s because it is – the Northern Ireland-only backstop was originally proposed by the EU.
As its name implies, it would see Northern Ireland remaining in the customs union on its own, rather than the UK as well.
So why is the arrangement so controversial?
The re-emergence of the Northern Ireland-only version is curious because Mrs May rejected it back in 2017.
This was because of concerns it would effectively create a border in the Irish Sea due to the need to check goods passing between Northern Ireland and Great Britain.
She pushed for a UK-wide version instead, in order to avoid this issue.
But either way, the Democratic Unionist Party and many Conservative MPs are opposed to the backstop.
They oppose it because they fear it will weaken the constitutional integrity of the Union – as Northern Ireland would be treated differently to the rest of the UK by following different rules and regulations.
The only difference between those two versions of the backstop is the degree to which this would happen.
Opponents have also expressed fears Britain could end up being trapped in the arrangement – and by extension the EU – indefinitely in some kind of “Hotel California” scenario – you can check out any time you like, but you can never leave.
They would only be satisfied by the complete removal of the backstop.
And it should be noted that numerous government figures have rejected the idea of pivoting back to a Northern Ireland-only backstop.
Business Secretary Andrea Leadsom told Sky News: “There isn’t a plan to have some sort of NI-only anything.”
DUP leader Arlene Foster held talks with Mr Johnson in Downing Street on Tuesday.
She said afterwards that the PM had “confirmed his rejection of the Northern Ireland-only backstop and his commitment to securing a deal which works for the entire United Kingdom as well as our neighbours in the Republic of Ireland”.
A third way?
Given the staunch opposition to the backstop, alternatives have been put forward and mooted.
According to Ireland’s EU commissioner Phil Hogan, the PM has suggested what he called an “all-Ireland food zone”.
In layman’s terms, this would mean agricultural goods and livestock having the same regulations across the island of Ireland.
This would build on checks already in place for livestock, but crucially would not cover tariffs on goods.
Mr Hogan told the Irish Times: “That is certainly a clear indication of divergence between Northern Ireland and the Republic of Ireland, the EU and the rest of the UK.
“This is the first time that this has been spoken about by a British prime minister where they are prepared to accept some level of divergence between Northern Ireland and the rest of the UK.
“If we can build on that we certainly might get closer to one another in terms of a possible outcome.”
But this might not fly with the EU, something that Mr Hogan hinted at in his interview.
“It would have to include all goods… in terms of any agreement,” he said, adding that he was hopeful “that the penny is finally dropping with the UK” on Brexit.