The Earth's inner core is seismically anisotropic, with the direction parallel to the rotation axis both fast and more attenuating. There is also increasing evidence that the inner core is asymmetric, with the western hemisphere exhibiting slower direction-averaged P wave velocity, less overall attenuation, and greater elastic anisotropy. It was recently suggested that the hemispherical variations might result from convective translation, whereby enhanced solidification in the western hemisphere leads to a net eastward translation of inner core material, with melting occurring in the east. Annealing accompanies this eastward movement. This study examines experimentally a previously unobserved sequence of grain growth and loss of texture during the annealing of a directionally solidified alloy. The growth of newly nucleated grains results because the original grains that resulted from directional solidification have a high energy associated with intragranular interphase boundaries, and because the minor element has a very low solubility in the primary phase so that a more traditional sequence of coarsening is not possible. This supplies a physical mechanism for the loss of texture that is suggested seismically.