Would an European mercantile company be able to found a country (or sovereign territory, rather) in the Caribbean or Oceania and have it run under private initiative?

Sorry, but your submission has been removed because we don't allow hypothetical questions. If possible, please feel free to rephrase the question so that it does not call for such speculation, and resubmit Otherwise, this sort of thing is better suited for /r/HistoryWhatIf.