Spherulites are birefringent structures with spherical symmetry, typically observed in crystallized polymers. We compute the band structure of opals made of close-packed assemblies of highly birefringent spherulites. We demonstrate that spherulitic birefringence of constituent spheres does not affect the symmetries of an opal, yet significantly affects the dispersion of eigenmodes, leading to new pseudogaps in sections of the band structure and, consequently, enhanced reflectivity.