The Caribbean islands in question (as pirate havens) were the Cayman Islands (not on our map) and most of the nominally "English" islands (the Bahamas, Antigua, and Jamaica are the classics). They were well known as pirate's havens, as the local (English) colonial governments were either unwilling or unable to stop the pirates, and thus the pirates used these as bases to prey upon other parts of the region.
Madagascar, in later years, was also known as a similar haven, as were the inland sounds on the North Carolina coast. I'm not familiar too much with Asian history, but I would not be surprised to learn that Taiwan played a similar role. NONE of these areas was ever actually governed by pirates, though - they were colonies of countries unable or unwilling to prevent the pirates from using them as bases.
In fact, it was quite common to wage economic warfare by licensing pirates; the deal was that the "licensed" pirates would not prey on their licensing country or its allies. These later came to be called "privateers", and were used as late as the early 1800s (including by the United States). The licenses were known as Letters of Marque and Reprisal, and I believe the USA Constitution specifically grants the USA government the power to issue them. (I know some were issued, but I'm not entirely sure it's in the Constitution.)
In game terms, however, representing the pirates as a country is a BAD idea. This would most likely prevent them from attacking except when at war, which (if fixed via events, etc.) would cause intense war exhaustion problems for EVERYONE, not just the pirates. This doesn't even get into issues like being able to eliminate piracy by conquering them, etc.