It is well known that the neutrino medium, such as the cosmic neutrino background, produces a tiny birefringence effect on electromagnetic waves. Recently, it was claimed that this effect may be enhanced by an additional presence of a plasma characterized by the electron plasma frequency. In our work, instead of considering the plasma, we consider a simple transparent refractive medium characterized by its index of refraction. To stay general we keep the mutual velocity of the refractive and neutrino media nonzero. For that situation we estimate the strength of the birefringence effect, analyse its dependence on the angle between the velocities of the two media, and compare the obtained results with other cases analysed previously.