  We study the propagation dynamics of bright optical vortex solitons in
nematic liquid crystals with a nonlocal reorientational nonlinear response. We
investigate the role of optical birefringence on the stability of these
solitons. In agreement with recent experimental observations, we show that
birefringence-induced astigmatism can eventually destabilize these vortex
solitons. However, for low and moderate birefringence, vortex solitons can
propagate stably over experimentally relevant distances.
