Belize actually is a third world country. Google it and look at the definition and what a third world country is. Belize is in that category. However yes, I agree, Belize is not "third world" like many other countries that you and I have been to.
I can't say for sure that there are NO Peace Corps volunteers on the island, but so far as I know they and all similar bodies withdrew from Belize at an official level a couple of years ago.
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