If you google the tests techniques you'll read about false positives from cross-reaction with related flaviviruses (e.g., dengue and yellow fever viruses)
In other words the antibodies of the persons history of diseases can make the test false positive.
I'm sure the person on Caye Caulker got the updated state of the art tests :-)
Florida , Texas and even Alaska have cases, so not coming to Belize to avoid Zika is still unrealistic fear IMHO.