The lowermost mantle beneath Central America has anisotropic seismic velocity structure manifested in shear wave splitting of signals from South American earthquakes recorded at North American broadband recording stations. Prior studies of deep mantle anisotropy in this region have characterized the structure as having vertical transverse isotropy (VTI), which is sufficient to explain a general trend of early tangential (SH) component arrivals. However, VTI models cannot quantitatively match systematic waveform complexities in the onset of many of the shear waves that graze this region. After accounting for splitting effects of upper mantle anisotropy beneath the recording stations, we model the corrected waveform data using full wave theory for mantle velocity models with an anisotropic D″ layer. This is the first attempt to quantitatively model a large data set including azimuthal anisotropy in D″. The models include transverse isotropy with either a vertical or tilted symmetry axis, the latter resulting in azimuthal anisotropy. For some initial shear wave polarizations, tilted transverse isotropy (TTI) produces small, reversed polarity arrivals on the SV components at the arrival time of SH, consistent with the data. Geographical variations in the azimuth of the TTI symmetry axis are indicated by the data. The lack of azimuthal coverage prevents unique resolution of the TTI orientation and also precludes distinguishing between TTI and other azimuthal anisotropy structures such as that predicted for lattice preferred orientation of minerals. Nonetheless, our modeling demonstrates the need for laterally varying anisotropic structure of more complex form than VTI for this region.