what’s the significance of the Carlist flag?
That structure is the San Felipe del Morro castle in the Old San Juan National Historic Site, which is run by the U.S. National Park Service. They still fly the flag to highlight the 400 year Spanish heritage of the complex, and as sort of reconciliation token for the island’s Spanish population. That was also the last flag of the Vice Royalty of New Spain, which was in effect when the island was ceded to the U.S.
They supported Franco? Might be the PR Version of the Gadsden Flag.