You know the ones, the little pieces that come from rear door window to rear windshield. Should I take them off? They're quite the nuisance to stick back and rattle like hell. Can I do without 'em?
Bob's right, that is the vinyl top trim. But your vinyl top is MIA, so I say remove it if it bothers you.
The vinyl top option was just a vinyl covered roof. The edges were dressed up with the trim that you are describing. So, someone before you has simply removed the vinyl covering. Which is probably why your trim is lose and rattles; the vinyl is not under it, it is simply gone.