This might be too late (I didn't notice this Meta post until after several half-lives of viewing activity had already elapsed), but I'll clarify the intent of my original comment. Here's a copy of the comment:
Tachyons are not consistent with mainstream physics, so the only way to answer this "what if" question is to use a non-mainstream theory. What theory do you want to use?
Instead of casting a close-vote, I posted this comment as a request for clarification. I'm guessing that the question intended to use Maxwell's equations but with a charge-current four-vector $J^\mu$ that is nonzero only along a spacelike worldline, but that's still not enough information. In order to answer the question about bound states, we also need to specify the equations of motion of the charged matter (maybe the relativistic Lorentz force equation but using proper length in place of proper time?), and we need to define what "bound state" means when the particle world-lines are spacelike. Neither of those is self-evident, at least not to me.
Regarding whether or not the question should have been closed as off-topic: In my opinion, a question may be on-topic for this site even if it's about a toy model, which is what physicists call a simplified/modified model that isn't necessarily a good approximation to anything real but that is used to explore some specific mathematical feature(s) of other theories that are good approximations. The study of toy models makes important contributions to the progress of mainstream physics, so in hindsight, I probably should have worded my comment differently, like this:
Tachyons are not consistent with our current understanding of nature, so the only way to answer this "what if" question is to specify a toy model. What toy model do you want to use?
Regarding why other questions about tachyons are still open: maybe many of them shouldn't be. I don't know. I haven't even read most of them.