Brexit is about drawing lines on maps and hearts. In that respect it is at odds with the 1998 Good Friday agreement which sought to erase them. Key to ending the Troubles was the removal of the hard border between the north and south of Ireland. Theresa May’s problem is that she has committed to leaving the European Union while respecting the peace deal that depended on both London and Dublin being part of it. The agreement made less salient the questions of identity: under its provisions people in Northern Ireland could be, uncontroversially, citizens of Britain or Ireland. What did it matter if they were part of a club that allowed people to travel, work and reside anywhere? North-south institutions were set up to oversee common endeavours. The agreement was sold as a “stepping stone to a united Ireland” to republicans, while unionists held that it was “securing the union”. The symbolism was key: with a hard border you can see that the island is partitioned, without it you cannot.

Aware of this, London and Dublin committed not to return to the days of border checkpoints, which would risk undermining the peace deal’s principles. Mrs May’s withdrawal agreement enshrines this in law in the form of an insurance policy: to maintain an open border on the island of Ireland in the event that the UK leaves the EU without securing an all-encompassing deal, there would be a backstop arrangement to allow for frictionless trade. Fanatical rightwing anti-EU Tory MPs were not bothered about the peace process, or the aspirations of fellow subjects. Instead, they saw in the insurance policy a devious mechanism to force Britain to march in lockstep with EU regulations that hard Brexiters were desperate to get out of. To reverse her government’s historic Commons defeat, Mrs May voted to replace the backstop. She did so to win over the Democratic Unionist party, but her actions enraged businesses in Northern Ireland which had publicly backed her deal.

In Belfast on Tuesday, the prime minister was back to the push-me-pull-you politics of Brexit. She signalled that, rather than the “alternative arrangements” to the backstop floated previously, she was edging towards an insurance policy that could either be time-limited or allow for a unilateral withdrawal. The prime minister may have said more than she intended, as a group of her MPs considering “alternative arrangements” were meeting in London as she spoke. Neither of Mrs May’s offers seem possible in a strictly legal sense, but both could be rendered in political terms. Almost everyone agrees that the backstop must be a temporary measure, but no one wants to say so in law. On such anvils, political follies can be forged. Mrs May’s problem stems from the fact that the only Northern Ireland party represented in the Commons is the DUP. The party’s role in itself undermines a core principle of the Good Friday agreement, that the position of the British government would be “neutral” – it had no selfish, strategic or economic interest in Northern Ireland and would accept the unification of Ireland by consent. The DUP-Conservative pact, a result of the 2017 election, calls this aspect into question. The DUP represents only a third of Northern Irish votes cast in 2017.

Even more glaring is that Northern Ireland voted to remain. Unionism risks defeating itself if it becomes too closely identified with Brexit. Belatedly, Mrs May’s ministers have woken up to the democratic deficit in Northern Ireland and called for the restoration of Stormont. These islands are sold as a story of stability. Political scientist Roger Awan-Scully says history would differ. Less than a century ago, the UK lost a greater proportion of its territory – through Irish independence – than Germany had lost in the treaty of Versailles. In 2014, close to half of Scotland voted for independence. Fragmentation is by no means inevitable, but without a sense of common purpose and community it becomes more and more a possibility.

• This article was amended on 6 February 2019 to change “The DUP represents only a third of votes cast in 2017” to “The DUP represents only a third of Northern Irish votes cast in 2017”.