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 As stated, though, this is already withdrawn.

Let's have a neutrino and a photon and assume that neutrinos transform under Galilei while photons transform under Lorentz. Adjust the impulse of the neutrino so that it moves exactly at v=c parallel to the photon. If they are fired towards the detector at the same time, they will be detected at the same time.

Now change the reference frame to one of an observer moving in the same direction at c/2 (or any other arbitrary velocity). With respect to this observer, the photon moves still at c according to Lorentz while the neutrino moves at c/2 according to Galilei. Therefore the photon will reach the detector before the neutrino does.

This is a paradox. Either the detection of the neutrino and the photon are one event or not, it cannot depend on reference frame.