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.
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.)