I can't believe no one mentioned Tropico: it fits perfectly OP's description.
I haven't played Tropico 6 which adds archipelago and online multiplayer, seems fine, my personal favourite is still Tropico 4 (you can get it for very cheap with all the DLCs during sales) even though it has a more colorful style than other entries. Tropico 3 is also fine, a bit more rudimentary, while 5 was a bit criticized for dumbing down some Tropico 4 features.
Honorable mention for the music that also adds a lot to those tropical / caribbean vibes