  We revisit the model building challenges that one faces when trying to
reconcile the OPERA claim of neutrino superluminality with other observational
constraints. The severity of the supernova bound and of the kinematical
constraints of Cohen-Glashow type lead us to focus on scenarios where all types
of particles are superluminal inside matter. In contrast to the Dvali-Vikman
proposal, this matter effect needs to be very short-ranged to avoid constraints
from experiments on the Earth's surface in low-density environments. Due to
this short range, the interaction underlying such a matter effect would have to
be far stronger than permitted by fifth-force bounds. As a conceivable way out
we suggest to make the matter effect "binary", i.e., dense matter does not
directly trigger superluminality, but merely induces the transition to a
different phase of some weakly coupled hidden sector. This phase exhibits
spontaneous Lorentz violation or at least a stronger than usual mediation of
some residual Lorentz violation to all matter. The effect has not been observed
before since we have never before been able to measure the velocity of
high-energy particles in dense matter with sufficient precision.
