LONDON — The militarized checkpoints that once stood along the border between Ireland and Northern Ireland became flash points for sectarian violence during the Troubles, and no one wants to see their return after Britain quits the European Union in 2019.

But a document released by the British government on Wednesday on how to preserve the open border there has underscored the sprawling complexity of Britain’s planned departure from the bloc, known as Brexit.

Northern Ireland is a part of the United Kingdom, while Ireland has been an independent nation since 1921. Both are members of the European Union — a shared status that has helped London and Dublin put aside historical differences and develop such a close relationship that border controls have disappeared. Travelers generally do not even know when they have passed from one country to the other.

After Brexit, however, the roughly 300-mile frontier with Ireland will be the United Kingdom’s only land border with a bloc whose economic arrangements, including its customs union and single market, it plans to leave. That creates a host of problems.

The customs union allows members to trade freely among themselves while charging a single tariff on some goods from nonmembers. When Britain leaves the bloc, goods crossing the border from Britain into Ireland could be subject to varying tariffs, unless the British adopt the same tariffs as the European Union or strike a special deal with it. Policing those varying tariffs could be burdensome.

The same principle holds for the single market. It is maintained through a complex and detailed set of standards that Britain would either have to abide by or face the logistical nightmare of checking goods entering the European Union from its territory.

Adopting the same tariffs and standards as the European Union would clear up a lot of problems, but would undermine the supposed purpose of Brexit in the first place, which is to re-establish control over immigration and national sovereignty. It would also complicate, or perhaps even preclude, forging trade deals with countries like the United States, another major goal of Brexit.

The document published on Wednesday represents the first, if somewhat vague, attempt to deal with these problems as they affect the Irish frontier. It rules out the reintroduction of physical infrastructure such as customs posts, and there appear to be no plans to use security cameras or license plate recognition technology at or around the border.

Immigration would not be policed at the Irish frontier, nor would there be passport checks on people entering mainland Britain from Northern Ireland.

That would seem to raise the possibility that European Union citizens could enter Britain indirectly through Ireland, perhaps undermining control over immigration. But the document hints that European citizens would probably be allowed to enter Britain freely and directly from Europe even after Brexit, though they might face some restrictions on their right to work or to claim welfare payments as people from outside the European Union do today.

The bigger problem is trade. The European Union’s chief Brexit negotiator, Michel Barnier, warned last month that “frictionless trade” would not be possible if the United Kingdom left the bloc’s economic arrangements.

Wednesday’s document calls on the European Union to agree to a series of waivers for small businesses and farmers, to avoid the need for them to complete customs formalities.

But that is only part of the problem. Even if that were agreed, larger companies would surely face higher costs. The British government is hazy on this point, talking about setting up simplified customs procedures and applying technologies — so far unspecified — to track goods, reduce bureaucracy and prevent costly delays.

British officials say there is so far no estimate of the increased cost that some businesses would face. Stephen Martin, the director general of the Institute of Directors, a business lobby group, described the document as a “significant step forward,” while adding that it “throws up even more questions about how much flexibility and imagination will be needed to overcome some very fundamental challenges.”

John Bruton, a former Irish prime minister, said the document failed to address the need for tariffs to be collected by Ireland on some goods imported into the European Union. “Brexit is going to increase the cost of doing business,” he told the BBC.

Farmers may have to adapt, too. More than 10,000 pigs are exported from Ireland to Northern Ireland every week, while a quarter of all milk produced by dairies in Northern Ireland is exported to Ireland for processing.

To minimize disruptions to trade, the document suggests setting common regulatory standards on agricultural products traded between the United Kingdom and the European Union. Yet that could complicate London’s efforts to strike global trade deals with countries such as the United States, if they involve importing agricultural goods.

Ireland’s government gave the document a polite but cautious reception, with its foreign minister, Simon Coveney, welcoming the principles behind its approach. Yet he added, according to the Irish broadcaster RTE, “What we don’t have, though, is the detail as to how it’s going to work.”

Mark Daly, a senior member of Ireland’s opposition Fianna Fail party, was less diplomatic, describing the plan as “pie in the sky” and warning that the proposals amounted to a “smugglers’ charter.”