Here’s an issue that will quickly occupy Brexit negotiators once they get down to business next month: the appearance soon of a new European Union border on the island of Ireland.

The frontier poses a real problem for both sides. Britain’s departure from the bloc, which the U.K. and Ireland joined in 1973, threatens the free movement of people and of goods across the border that comes with EU membership. According to Irish government figures, 177,000 trucks, more than 200,000 vans and 1.85 million cars take advantage of this every month over some 200 crossing points.

Both the U.K. and Ireland have said they want no return of a “hard border” and the barriers and border posts reminiscent of the conflict in Northern Ireland that killed more than 3,600 people over three decades. “You can see how fast we come to a question of war and peace,” German Chancellor Angela Merkel said Wednesday in remarks about Brexit and Ireland.

In an interview last week, Tony Blair, the former British prime minister, said negotiators “should treat Ireland as a special case and we should do everything we possibly can to preserve what is there now.” Mr. Blair was an architect of the accords that brought peace to Ireland, which even after 20 years remains fragile.

He said he detected “a huge desire in the U.K., in Ireland and in Europe to come up with a smart solution.”

But current Prime Minister Theresa May made two pledges in her Conservative Party election manifesto Thursday that made a hard border more likely.

She promised that Britain would leave the EU customs union, which means customs checks on goods, and to cut net migration to the U.K. to tens of thousands annually from hundreds of thousands today, which implies identity checks on people.

Denis MacShane, a Europe minister when Mr. Blair was prime minister, said Irish politicians and others who insisted there will be no customs checks and border posts between Northern Ireland and the Republic of Ireland if Britain leaves the customs union “are pulling the duvet over their heads.”

As Mr. Blair pointed out, leaving the customs union and the EU’s zone of common regulation known as the single market is an issue mostly for the EU. The bloc will seek to ensure that goods not conforming to EU standards don’t pour over the border from the U.K. via Ireland into the EU.

On the other hand, it is the U.K. that wants control over movement of people.

But the U.K. has also said it wants to preserve the so-called Common Travel Area between the U.K. and Ireland that dates back to 1923 and allows citizens of each country to travel to and live and work in the other.

So how does this all happen? One former senior British civil servant said this week: “Everybody knows where we want to end up but we don’t know how to get there.”

There are three approaches that could help.

Technological. Electronic customs checks operate widely across the world. Ireland’s Revenue Commissioners say 92% of goods imported into Ireland from countries outside the EU are processed and cleared within 30 seconds. Just 2% of shipments are physically examined. Ireland will have to scale up its operation to cope with U.K. imports.

Customs clearance isn’t necessary now on trade with the U.K. but, after Brexit, many checks could also be carried out at approved premises of importers in the North, away from the frontier, the commissioners say.

Spot checks and big fines. This would allow people to self-declare what they are bringing into the EU, even via an app on their smartphones. There could be border monitoring of vehicles and spot checks together with hefty fines for violators.

Political. The entire island of Ireland could remain part of the single market and the customs union. In practical terms that would push the customs clearance problem over the Irish Sea into Great Britain. The Common Travel Area could be retained with the U.K. able to carry out identity checks between the island of Ireland and Great Britain. This happened during World War II, when Ireland was a neutral country.

The big problem with that would lie in the likely fierce opposition from Protestants in Northern Ireland loyal to the British crown. They would probably view it as the start of a slippery slope toward the political unification of Ireland.

The smoothest solution is thus almost certainly ruled out on political grounds. The other ideas reduce the need for barriers and guards at border crossings. But they couldn’t replicate today’s frictionless movement of goods and people.