The usual description of self-induced flavor conversions for neutrinos (ν’s) in supernovae is based on the simplified assumption that all the ν’s of the different species are emitted “half-isotropically” by a common neutrinosphere, in analogy to a blackbody emission. However, realistic supernova simulations show that ν angular distributions at decoupling are far from being half-isotropic and, above all, are flavor dependent. We show that flavor-dependent angular distributions may lead to crossing points in the angular spectra of different ν species (where Fνe=Fνx and F¯νe=F¯νx) around which a new multiangle instability can develop. To characterize this effect, we carry out a linearized flavor stability analysis for different supernova neutrino angular distributions. We find that this instability can shift the onset of the flavor conversions toward low radii and produce a smearing of the splitting features found with trivial ν emission models. As a result the spectral differences among ν’s of different flavors could be strongly reduced.