MA has the same system. An Independent has to declare which ballot (Rep,Dem) s/he wants before s/he can vote.
I think the problem is that you're not allowed to change your affiliation at your polling place. The story I heard was that the person was registered as Independent. He was told by poll workers that he had to declare himself to be either R or D....Ind. can't vote. HE even questioned being able to do that. Another poll worker said they're not suppose to but they let him anyway. So Florida doesn't have the same system...they actually have rules against it but apparently those rules went out the window.