How do we know tachyons don't exist? 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?
 A: 
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?

The objection is from mainstream physics, and main stream physics is about mathematical models that fit observations and are predictive of future observations.
Laws of physics are the axioms necessary to pick up the mathematical functions relevant to measurements and observations.

A tachyon /ˈtæki.ɒn/ or tachyonic particle is a hypothetical particle that always moves faster than light. Most physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such particles did exist, they could be used to build a tachyonic antitelephone and send signals faster than light, which (according to special relativity) would lead to violations of causality.

Italics mine.
We have not observed or measured in our laboratory experimentally  violations of causality, i.e. effects before cause, or communications from the future.
(If the mediums' communications  , messages from the future,  become accessible to laboratory experiments, maybe a drastic revision of the laws of physics will allow tachyonic particles in our list of observable particles.)
