[greymule]

First, you should make the call by what you see, not by some notion of "what everybody else is seeing."

That said, the play you describe is something of a HTBT. If the runner obviously beat the throw but slid completely past the plate without touching it, then you are now actually dealing with a missed-base situation, which requires an appeal the same as if a runner beat the throw to 1B but missed the bag. In ASA, before the runner returns to touch the plate, F2 can step on home and appeal to you, or (more likely) tag the runner. But if the runner had actually slid past home plate before the ball arrived, the call is safe, even if she missed it, until there's an appeal.

On the other hand, if everything happened practically simultaneously—the runner is reaching for the plate while sliding past the bag (e.g., "shoulder level with the plate), then I'd call the out. And what the fans might think is irrelevant. Incidentally, even if you were concerned with what the fans think they saw, it should have been obvious to everyone that the slap of the plate was an indication of an original miss. An experienced umpire might sell the call by announcing, "Out! Before the hand got in."

[Note: the immediate appeal by stepping on the plate with the runner still in the vicinity is possible in ASA but not some other codes. I don't know how Fed treats that play, for example.]