A coincidence, however, it is not. Technically, neither Mr. Trump nor the Trump Organization owns the property (or most of the other buildings featuring the Trump name outside the United States). But in 2010, Mr. Trump allowed the building’s Turkish owners, Dogan Holding, to brand it with his name, in exchange for a sizable fee. The total has not been disclosed, but campaign records show that by July 2015, Dogan Holding had paid Mr. Trump between $1 million and $5 million for the use of his name.

It was the revelation of that deal that had the Sumeli sisters making for a premature exit. “Why should I respect a president who doesn’t respect my veil?” asked the younger Ms. Sumeli, who is studying child development. “We won’t be coming here again,” her elder sister added.

The sisters were among several visitors to take issue with Mr. Trump’s attempts to suspend immigration from seven Muslim-majority countries. While Turks were not subject to the ban, most interviewed at Trump Towers were nevertheless offended by its principle and expressed solidarity with those from the affected countries.

“It doesn’t matter whether Turkey is included or not — I’m against it,” the younger Ms. Sumeli said while gathering her things. If “my government banned Christians,” she added, “I’d be against that, too.”

Trump Towers Istanbul is a two-pronged construct: an office block and an apartment complex that jut skyward from a multistory mall at the bottom.