  We provide a vectorial model to simulate second-harmonic generation (SHG) in
birefringent, transparent media with an arbitrary configuration of non-linear
($\chi^{(2)}$) crystalline grains. We apply this model on disordered assemblies
of LiNbO$_3$ and BaTiO$_3$ grains to identify the influence of the
birefringence on the random quasi-phase-matching process. We show that in
monodispersed assemblies, the birefringence relaxes the grain-size dependence
of the SHG efficiency. In polydispersed assemblies with sufficiently large
grains, we find that the birefringence introduces an SHG efficiency enhancement
of up to 54% compared to isotropic reference crystals, which is grain size
independent. This enhancement increases linearly with the grain size, if the
birefringent grains can be phase matched. These two different scaling
behaviours are used in Kurtz and Perry's powder-technique to identify the
phase-matchability of a material. We show on the example of LiNbO$_3$ and ADP
that this technique cannot be applied when the grains get smaller than the
coherence length, because the SHG scaling with the grain size becomes material
specific.
