Superluminal particles with causality What kind of CLASSICAL theories would allow to true (non-apparent) superluminal particles (beyond speed of light, BSOL) agreeing with causality to exist? I mean, are causal superluminal classical objects forbidden in any theory? Please, note that the "prejudices"/hypothesis for this question are: 
1st. Classicality ansatz. A classical theory is defined as a theory containing (likely) SR as (low energy, or some other parameter) limit. I don't restrict it to usual Lorentz invariance. So, perhaps, the term "classical" should be complemented with classical invariance (group).  
2nd. Causality hypothesis. Causality in the usual physical sense, i.e., every effect is preceded by some cause. 
3rd. Superluminality. BSOL "definition": a superluminal particle is any object who can travel at higher speed than (known) light "in vacuum", i.e. $c$, without violating the previous two hypothesis, namely, "classicality" and "causality". 
So my question can be also formulated as follows: can we build a "sensible" physical theory where the above 3 postulates do hold? 
In the context of 3+1 relativity, with c as invariant velocity it is imposible. 
But is every theory with true superluminality necessarily a causality-violating (CV) theory? 
 A: Superluminal particles (we call them elvisebrions) are possible if a hidden sector exists which is either not Lorentz invariant or it is Lorentz invariant but with a different limiting speed. We discuss this possibility in a rather entertaining manner in https://arxiv.org/abs/2107.10739 (Relativity 4-ever?).
By the way, the tachyon hypothesis was first suggested much earlier by Lev Yakovlevich Shtrum in 1923 (see the above paper and references therein).
Zurab Silagadze.
A: Nice question.
I don't understand the Lorentz-violating possibilities very well, so I'll only try to comment on Lorentz-invariant theories.
The classic papers are Tolman 1917, Bilaniuk 1962, and Bilaniuk 1969. Bilaniuk 1969 can easily be found online by googling, and gives a good overview. Tolman proposed a causality paradox involving tachyons, known as Tolman's antitelephone. Bilaniuk et al. claimed to resolve the causality issues, basically by saying that when we do a Lorentz transformation that results in a tachyon going backward in time, we reinterpret it as an antiparticle going forward in time. Some people today seem to believe that this resolved the paradoxes, but the majority view seems to be that it didn't. I don't believe it resolves them. Their description involves processes such as a tachyon whose world-line is a line segment connecting events A and B, and for which observers at both A and B believe that they transmitted the tachyon rather than receiving it. To me this seems like a clear violation of causality, and it seems like Bilaniuk has done nothing more than relabel some of the events involved in the paradox.
It's possible to have two tachyons with their four-momenta chosen such that the total four-momentum is zero. This means that any theory involving tachyons allows pairs of them to spontaneously appear or disappear. In a classical theory, it seems hard to reconcile these spontaneous events with causality. In the Bilaniuk interpretation, rates of emission in one frame correspond to rates of absorption in another frame. Again, this is hard to reconcile with causality.
A more modern point of view is given by Baez. The wave equation for a tachyonic field has real-energy solutions plus imaginary-energy solutions that blow up exponentially. The exponential solutions are clearly unphysical, but if you exclude them, you don't get uniqueness and existence of solutions to Cauchy problems. In my view, uniqueness and existence of solutions to Cauchy problems is a good definition of causality.
Baez, http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html
Bilaniuk, Deshpande, and Sudarshan, '"Meta" Relativity,' Am J Phys 30 (1962) 718
Bilaniuk and Sudarshan, "Particles beyond the light barrier," Phys. Today 22, 43 (1969)
R. C. Tolman, The Theory of Relativity of Motion (Berkeley, 1917), p. 54
A: About faster than light...
I know (in fact I am currently yet studying) different extensions of relativity. Some options naturally arise:
1) Yes, Ben... Sudarshan's (and Recami's) Meta-relativity  is one "option", somewhat oldfashioned. Problems: tachyons have not been observed in Nature yet.
metarelativity paper
metarelativity paper 2012
2) Carlos Castro's/Matej Pavsic extended relativity in C-spaces gives another posibility. You can have different kinds of velocity in C-space. So, the trick used there to avoid the speed of light limit is to add "extra degrees of freedom" living in the C-space. Note that it also keeps the notion of invariant relativity! Problems: we have not discovered apparently any experimental hint of C-space. Relativity in C-spaces: review
3) I discussed another option recently in my blog entry, something quite unknown for many physicists (or neglected) and something that sci-fi writers don't understand. Hypertime. If you have "new" timelike dimensions and different speeds of light, or even if you have multiple times and the same invariant velocity, you can modify the speed of light limit. It is the hypertime and not the hyperspace what makes faster than light motion possible. Note that you can even keep a notion of invariant speed. crystalline relativity 
Problem: apparently, Lorentz symmetry holds yet in any experiment, so Kalitzinian metrics (semiriemannian metrics) have not appeared in Nature yet (even the hypothesis of the spacetime quasy-crystal is crazy, but Wilczek himself or Petr Jizba have proposed similar ideas...)
i) Time crystals ii)Time crystals II iii)World crystal
4) The general relativity trick via wormholes (i.e., non trivial topological connections between two points in spacetime) or via Alcubierre warp-drives. Problem: known quantum instabilities and weak energy condition violations. 
wormholes
alcubierre drive
Some time ago, even something like the Weak Energy Condition was critized (I think there is no such an opposition to WEC now), but I believe the main problem is of course the quantum theory (something I gave up in the original question and that would deserve additional research/thread for any of these 4 options or answers). 
What about classicality? With careful analysis, the 4 options above can be considered "classical". 
What about causality? Nobody understand what time is and a change in our notion of fundamental symmetry and what means "a clock" should avoid the causality problems. In fact, with multiple time-like dimensions, it is hard to assest if what causality violations cause in 1d time could happen in nd time...And I am not sure how multi-time theories can avoid causality, but I think it could be possible. Note that there have been studies of mechanics with multiple times in the recent literature. 
