1) if neutrinos can beat lightspeed, relativity could simply not apply to them, rather than applying in some weird twisted way.
2) Neutrinos are nearly perfectly exempt from photon interactions (the simplest diagram is to pop out a virtual W and electron).
3) This could be the reason for 1.
4) How, then, do other charged particles get all relativized? Well, photons get entangled with just about everything just by virtue of most things having electrical charge. That's what the renormalization theory is about - including all of the recursing interaction effects that a particle can't get rid of into its basic description.
So, if you have, say, innate Galileian relativity for most particles but not for light, which has a constant speed in a preferred rest frame, charged particles are going to act at least a little bit like relativity predicts - as the particle approaches light speed, the bow wave part of its E-field gets stronger and stronger, accumulating energy... as it approaches the speed of light the photon energy would diverge like on a jet about to go supersonic, except that unlike the jet, you simply can't get the photons out of the way fast enough.
Hmm. That really doesn't work, because it suggests that the mass dilation would vary proportional to charge, not to mass. So I withdraw the first notion too.
If Lorentzian relativity doesn't work, Galilean relativity is certainly not what replaces it. Moreover, I find it difficult to imagine how you could have photons transforming under Lorentz group and other particles transforming under Galilei (or whatever else) group. This will certainly produce inconsistencies, unless you managed to explain them away in some clever complicated way.
I would rather accept that neutrinos are indeed tachyons, whatever weird consequences it may have.
Link: nextbigfuture.com/2011/11/faster-than-light-neutrinos-opera.html
Link: science20.com/quantum_diaries_survivor/opera_confirms_neutrinos_travel_faster_light-84763
Paper: kruel.co/paper-neutrino-velocity-JHEP.pdf
Previously on LW: lesswrong.com/lw/7rc/particles_break_lightspeed_limit/