A persistent reversal in the expected polarity of the initiation of vertically polarized shear waves that graze the D'' layer (the layer at the boundary between the outer core and the lower mantle of Earth) in some regions starts at the arrival time of horizontally polarized shear waves. Full waveform modeling of the split shear waves for paths beneath the Caribbean requires azimuthal anisotropy at the base of the mantle. Models with laterally coherent patterns of transverse isotropy with the hexagonal symmetry axis of the mineral phases tilted from the vertical by as much as 20 degrees are consistent with the data. Small-scale convection cells within the mantle above the D'' layer may cause the observed variations by inducing laterally variable crystallographic or shape-preferred orientation in minerals in the D'' layer.