Since the 15th century, southern Bangladesh’s “mosque city” of Bagerhat, dominated by the imposing brick Sixty Dome Mosque, has been a pilgrimage destination for pious Bangladeshis and tourists.

“I believe if I wish something and visit the mosque, Allah will accept my wish and I’ll see success in the future,” said Shofik Ahamed, a university student who explored the landmark, a World Heritage Site, on his recent Eid holiday.

But Sixty Dome Mosque and a sprawl of dozens of other medieval mosques, public buildings, tombs and graveyards at the estuary confluence of the Ganges and Brahmaputra rivers face an increasingly uncertain future as the planet warms...