PLATEAU ROSA, Italy—Or is it Switzerland?

Running a restaurant high in the Alps is hard enough. Lucio Trucco also has to figure out which country he’s serving spaghetti in. Melting glaciers are changing the shape of Alpine crests, and with them, international borders.

When Mr. Trucco’s restaurant and lodge, the Rifugio Guide del Cervino, opened in 1984, the rocky promontory on which it sits was indisputably in Italy. But in the past 15 years, the surrounding glacial ridge that defines the national boundary has moved decisively, leaving the building on the Swiss side. Or so says Switzerland.

Mr. Trucco gives that notion a frosty reaction. “The building hasn’t moved, so this is still Italy,” says the 49-year-old mountain guide as he stands on the building’s balcony, with majestic views of the Matterhorn to his right and Italian peaks to his left.

As further evidence he cites the menu, which features spaghetti alla carbonara and tagliatelle con porcini. “There aren’t any Swiss dishes! In Switzerland, you buy chocolate and watches. In Italy, you come to eat pasta.”

At high altitude, where thick mantles of ice and snow cover rocky crags, a legal convention between Italy and Switzerland defines their frontier along the “watershed,” or the dividing line where glaciers slide slowly downhill toward one country’s valleys or another’s.

If the watershed moves, so does the border. Italian and Swiss officials occasionally ascend the mountains armed with GPS devices, satellite photos, maps and drones to record where exactly their frontier stands.
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Italy’s border with Austria also follows the glacial ridges. France refused to sign up to a similar arrangement, because after World War II it gained territory on the Italian side of Alpine watersheds and it doesn’t want to lose ground.

Until recently, mountain ridges didn’t move much. Switzerland might expand a little here, while Italy grew a little there. With only small patches of snow at stake, both countries were chill.

The warming climate is steadily shrinking glaciers, often in a lopsided way. Flanks that face south, or that have thinner snow cover due to wind exposure, can lose mass faster, says Dr. Frank Paul, a senior glacier researcher at Zurich University. As a result, a glacier’s ridge or highest point can drift sideways.

The big acceleration started in 2003, says Mr. Trucco, when a heat wave swept through Europe and led to ice melting in the Alps at higher altitudes than usual. Since then, Europe has had many of its warmest years on record, and glaciers have continued to melt.

About half of the glacier volume in the Alps could be lost by 2050, according to a recent study by the European Geosciences Union, a research institute. That could create more high-altitude headaches for Swiss and Italian officialdom.

Until the mid-2000s, ice and snow on the Plateau Rosa formed a ridge some 15 yards east of Mr. Trucco’s restaurant, blocking the view down into Switzerland, says Mr. Trucco. That ridge has disappeared—bringing into view a ski lift in Switzerland—and now the highest point on the plateau is farther west, toward the Italian ski resort of Breuil-Cervinia.

The Swiss say the building, which also has communal rooms with about 40 beds, is in Switzerland. It made the claim in 2013 following an on-site inspection. Italy refuses to recognize the border change.

The restaurant sits about 11,500 feet above sea level, high enough that some of the nearby slopes are open in the summer. On a recent morning, skiers were taking a break inside, drinking coffee and eating cake while Italian pop music blared from speakers outside. A menu written on a chalkboard announces the offerings and a wood-burning heater warms the cozy seating area.

If the building is acknowledged to be on Swiss territory, then all the food and other products brought up by cable car from Italy’s Breuil-Cervinia would potentially need to pass customs inspections.

The building would also become subject to Swiss law and taxes. Everything from its electrical sockets to the layout of the kitchen would need to be changed to meet Swiss regulations.

It’s an existential problem for Mr. Trucco. He wants to do renovations, but can’t until he knows which country to ask for various building permits.

A few years ago, Alain Wicht of Switzerland’s Federal Office of Topography, known as Swisstopo, met his Italian counterparts at the Rifugio del Cervino to try to unfreeze the situation. A compromise was floated: Italy could keep the building in return for territory elsewhere.

“We had a solution,” says Mr. Wicht. “We thought everybody was happy. But when it was time to sign, the Italians backed out.”

The Italian army, whose Military Geographic Institute defends the country’s map outline, declined to comment. Fresh negotiations with the Swiss are due this spring.

Mr. Trucco doesn’t hold out much hope for a thaw. Standing astride the glacial ridge on the Plateau Rosa, with one foot on the Italian side and one on the Swiss, he outlines an idea to both promote his restaurant and end the cold war.

“My solution as a businessman is for the building to be divided between the two countries,” he said. “Let’s use this situation to our advantage. We can have tables that are half in Italy and half in Switzerland. People will love that they can eat with somebody who’s in a different country.”

Some of his clients sound skeptical. “Switzerland is over there,” said skier Lorenzo Nigro, a businessman from Florence, gesturing toward the Matterhorn.

Even some Swiss would like this spot to stay abroad. The Rifugio del Cervino offers them a quick visit to Italy without having to leave the Swiss ski area of Zermatt.

“I might still come here if this were Switzerland, but I don’t think it would be the same,” said Stefan Burchard, a student from Zurich. “And I don’t think the lasagne would be as good.”