We investigate the polarization characteristics of apophyllite crystals in an attempt to evaluate their potential use for achromatic waveplates. Among the 50 plates that we extracted and polished from natural apophyllite crystals, a few show the sought-for characteristic of birefringence that increases linearly with the wavelength. However, we also find that the crystals among our samples exhibit a sectored structure in their polarization properties, as well as an undesirable degree of spatial nonuniformity.