  Based on the analogue spacetime programme, and many other ideas currently
mooted in "quantum gravity", there is considerable ongoing speculation that the
usual pseudo-Riemannian (Lorentzian) manifolds of general relativity might
eventually be modified at short distances. Two specific modifications that are
often advocated are the adoption of Finsler geometries (or more specifically,
pseudo-Finsler spacetimes) and the possibility of birefringence (or more
generally, multi-refringence). We have investigated the possibility of whether
it is possible to usefully and cleanly deal with these two possibilities
simultaneously. That is, given two (or more) "signal cones": Is it possible to
naturally and intuitively construct a "unified" pseudo-Finsler spacetime such
that the pseudo-Finsler metric is null on these "signal cones", but has no
other zeros or singularities? Our results are much less encouraging than we had
originally hoped, and suggest that while pseudo-Finsler spacetimes are
certainly useful constructs, it is physically more appropriate to think of
physics as taking place in a single topological manifold that carries several
distinct pseudo-Finsler metrics, one for each polarization mode.
