What replaces the backstop?

Mr. Johnson’s plan — unnamed, as yet — would pull the whole United Kingdom out of the European customs union. That was a big victory for Brexiteers, opening a path to Britain striking its own trade deals, including with the United States.

It is also supposed to allay unionists in Northern Ireland because that region would legally be part of the United Kingdom’s customs territory, allowing it to retain its close ties to Britain and to benefit from future trade deals.

But the only way to avoid checks on goods passing from Northern Ireland into the Republic of Ireland was for the north to apply the European Union’s rules and procedures on tariffs — the same ones the Republic of Ireland follows. Mr. Johnson’s deal also keeps Northern Ireland aligned with European single market rules on industrial products and agricultural goods.

In practice, that meant that, rather than putting a border on the island of Ireland, Britain would have to put one in the Irish Sea, and impose regulatory and customs checks for items passing from Britain into Northern Ireland. Some analysts said that would create a separation of sorts between different parts of the United Kingdom, a proposal that Mrs. May said no prime minister could countenance.

Mr. Johnson has tried to mitigate the effects of that Irish Sea border with a system of rebates that would counteract higher European Union tariffs. Some goods, like personal items being carried by someone moving between Britain and Northern Ireland, would also be exempt from tariffs.

So far, though, this new system of checks has proved too much to bear for unionist lawmakers from Northern Ireland, who have refused Mr. Johnson’s deal and could ultimately be its undoing.