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.