The main mineral in the lower mantle, magnesium-silicate perovskite, transforms into a high-pressure, post-perovskite, phase at pressures and temperatures corresponding to the D'' seismic discontinuity approximately 200 km above the core-mantle boundary. The strong elastic anisotropy of post-perovskite has been invoked to explain the observed seismic anisotropy and to infer flow in the D'' region, based on models of textured post-perovskite. Such inferences rely on a knowledge of the mechanisms by which the post-perovskite can obtain texture. It is generally thought that seismic anisotropy in D'' is produced from lattice-preferred orientation generated during plastic deformation; however, it is difficult to explain all of the observed seismic anisotropy in D'' using a single deformation mechanism in post-perovskite. Here we show that strong texture inheritance is possible during transformation from perovskite to post-perovskite using a recently developed fluoride analogue system. If a similar transformation mechanism operates in the Earth, post-perovskite will inherit textures from deformed perovskite and vice versa as lower-mantle material passes into and out of regions of post-perovskite stability. This texture inheritance during the transition from post-perovskite to perovskite, combined with a single slip system in post-perovskite, can explain the seismic anisotropy of the lowermost mantle.