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.
It only produces an inconsistency if you treat either one as fundamental and try to base the other one off of it. As stated, though, this is already withdrawn.
Edited to add: An explanation on the downvotes would be appreciated. I was wrong. I said so. Yet the post with the errors is sitting at zero, and the post explaining why I even ever made the error, and this one, are downvoted. Seems sort of weird.
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/