While both Britain and the European Union agree that imposing a hard Irish border would threaten a very hard-won peace on the island, May insisted that there was no way she could allow anyone to "divide the United Kingdom into two customs territories".
Salzburg was an "informal" meeting rather than a full European Council summit, so there is no formal conclusion from the talks. "We are not ready to compromise on our four freedoms, on our single market as well as on the Irish borders", he insisted.
Challenges to her Brexit plan and her leadership will take center stage at a raucous party conference in early October.
'Everyone shared the view that, while there are positive elements in the Chequers proposal, the suggested framework for economic co-operation will not work'.
"Neither side can demand the unacceptable of the other, such as an external customs border between different parts of the United Kingdom - which no other country would accept if they were in the same situation - or the UK seeking the rights of European Union membership without the obligations", she wrote.
May has claimed that her proposals were the "only serious, credible" way to avoid a hard border in Northern Ireland.More news: Roger Federer Says Serena Williams Umpire Row ‘Really Interesting Case To Study’
As EU leaders met for the "family photo", a Reuters photograph caught the PM looking isolated from the rest - capturing the mood precisely.
The EU were clear that the backstop needed to be in the withdrawal agreement as a condition of a deal, but it felt it had demonstrated it was willing to compromise to address the political sensitivities in Britain.
"It was a good and fearless step by the prime minister", Macron told reporters.
France's president Emmanuel Macron said the EU's backstop should be "precisely preserved" in any agreement. Neither side wants an arrangement that would require the rebuilding of border infrastructure, the removal of which was a key part of the Good Friday Agreement that brought peace to Northern Ireland after years of sectarian strife.
If it left without a deal, the country would move from seamless trade with the European Union to customs arrangements set by the World Trade Organization for external states.More news: Trump on Kavanaugh: 'This is not a man who deserves this'
May faces opposition from not only a hard-bargaining Europe, which does not want to make it too appealing for members to leave their club, but also her own Conservative Party, populated by hard-line Brexiteers such as former foreign secretary Boris Johnson, who disparages the prime minister's exit plan as a weak capitulation to Brussels.
Macron acknowledged that Brexit was the choice of the British people but added that it was "pushed by those who predicted easy solutions".
If that happens, there would be 21 months of what is being a called a "transition period".
The First Minister said: "A no-deal Brexit will, by the United Kingdom government's own admission, lead to dire economic consequences and a shortage of medicines and foodstuffs".
However, at the gathering in Austria, Mrs May insisted there would be no delay to the UK's March 2019 departure, no second referendum and therefore the onus was on the continent's leaders to find a solution if they wanted to avoid a no-deal Brexit.
A new summit is expected to be set up for November, as the deadline for a deal by the end of autumn looms.More news: Hannah Gadsby’s Emmy speech a hilarious hit