Cheeses native to Caribbean?
I don't know if they have any in your area. But in Toronto you can get what I call can cheese at the local West Indian store. It is made on different islands, but I always get Tastee which is a brand made in Jamaica. I am not sure if you want to find it though, this stuff is like velveeta, but it doesn't melt (easily)
The search on Tastee cheese turns up a number web sites. Apparently this is a processed cheddar, probably using imported cheese from some where like New Zealand. Canning makes a lot of sense, given the hot climate and limited refrigeration (in the past). Same reason that Key Lime pie is made with condensed milk.
Looks like most of this cheese is eaten with 'Easter bun'
The Jamaica company: