Some bars ask if they can hold onto your credit card so that you can "run a tab". Im not fond of that idea. My card has been compromised twice on AC. Once for aprox $30 suspiciously from what was a popular beach front restaurant that is no longer in business. I dont believe the owners of the restaurant were responsible. The owner called and spoke to me. Employees will do crazy stuff. And the last time for about $2,000 in Air Transat airline tickets. I view the charges daily online and reported the bogus charges immediately. Nice thing about credit cards is that the credit card company assumes most if not all of the risk.
Belize does not have a monopoly on this activity. The same has happened to me after trips to Mexico and probably would be likely to happen in any other touristy location. Know what you charged and watch the transactions as they come through online.