In this article, we present a comprehensive, full band theoretical study of the high field, hole transport properties of the 4H phase of silicon carbide (4H-SiC). The calculations are performed using a full band ensemble Monte Carlo simulation that includes numerically tabulated impact ionization rates, and phonon and ionized impurity scattering rates. In addition, the simulation includes a mechanism, interband tunneling, by which the holes can move between bands in the proximity of band intersection points, It is found that there exists a significant anisotropy in the calculated steady-state hole drift velocity for fields applied parallel and perpendicular to the c-axis direction. Good agreement with experimental measurements of the hole initiated impact ionization coefficient for fields applied along the c axis is obtained, provided that interband tunneling in the proximity of band intersections is included in the model. If interband tunneling is not included, the calculated ionization coefficients are orders of magnitude lower than the experimental measurements.