Near-surface deformation between the Australian and Pacific plates in South Island, New Zealand is concentrated in a narrow zone marked by the Alpine Fault, but strong and widespread anisotropy inferred across New Zealand from shear wave splitting suggests diffuse deformation at mantle depths. To constrain this interpretation, we calculate temperature- and stress-dependent strain fields, crystal preferred orientation, anisotropy and resultant shear-wave splitting beneath a lithospheric fault deforming by either pure strike-slip or transpression. In pure strike-slip experiments, strain localizes in the high temperature regions under the thick continental landmass. Under the oceanic regions, which have thin crust, anisotropy is weak (delay times (dt) <0.5 s for a 100 km thick lithosphere). Under the continents, dt increases to 1.0 s for 100 km and 2.3 s for 200 km thick lithosphere. Dt saturates before 15 My (525 km displacement), at which point the polarizations of the first arriving shear waves (ϕ) are 25° from fault-parallel, similar to measurements in southern South Island. Further strain does not increase dt, but ϕ rotates to 15° from fault-parallel for displacements >1400 km. In transpression experiments that have initial structure based on estimates of the present crustal thickness and temperature, the cold root inhibits strain beneath the fault and leads to anisotropy patterns that do not explain the observations. This suggests that (a) a weak zone in the mantle lithosphere is needed to explain present-day deformation; and (b) most of the anisotropy measured may be "frozen in" from strike-slip deformation that occurred before the present mantle root developed.