  The usual description of self-induced flavor conversions for neutrinos
(\nu's) in supernovae is based on the simplified assumption that all the \nu'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 \nu 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 \nu species (where F_{\nu_e}=F_{\nu_x} and
F_{{\bar\nu}_e}=F_{{\bar\nu}_x}) around which a new multi-angle instability can
develop. To characterize this effect, we carry out a linearized flavor
stability analysis for different SN 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
$\nu$ emission models. As a result the spectral differences among \nu's of
different flavors could be strongly reduced.
