As I mentioned yesterday, Hollywood screenwriter working on a TV pilot about physics trying to get the details right.

What empirical evidence is there that tachyons do not exist? I understand that objects with mass cannot accelerate to (much less past) $c$. So anything capable of FTL travel would have to be massless or very strange. But is there any astronomical evidence that allows us to conclude that superluminal travel does not happen in nature? 

Like is there some specific phenomenon we would expect to see in the sky if non-free, interacting tachyons existed, and we're not seeing it? Or is the objection entirely mathematical?