  We show that fore-aft asymmetry, a generic feature of living organisms and
some active matter systems, can have a strong influence on the collective
properties of even the simplest flocking models. Specifically, an arbitrarily
weak asymmetry favoring front neighbors changes qualitatively the phase diagram
of the Vicsek model. A region where many sharp traveling band solutions coexist
is present at low noise strength, below the Toner-Tu liquid, at odds with the
phase-separation scenario well describing the usual isotropic model. Inside
this region, a `banded liquid' phase with algebraic density distribution
coexists with band solutions. Linear stability analysis at the hydrodynamic
level suggests that these results are generic and not specific to the Vicsek
model.
