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Charge carrier mobility is at the core of semiconductor materials and devices optimization, and Hall measurement is one of the most important techniques for its characterization. The Hall factor, defined as the ratio between Hall and drift mobilities, is of particular importance. Here we study the effect of anisotropy by computing the drift and Hall mobility tensors of a technologically important wide-band-gap semiconductor, 4H-silicon carbide (4H-SiC) from first principles. With $GW$ electronic structure and \textit{ab initio} electron-phonon interactions, we solve the Boltzmann transport equation without fitting parameters. The calculated electron and hole mobilities agree with experimental data. The electron Hall factor strongly depends on the direction of external magnetic field $\mathbf{B}$, and the hole Hall factor exhibits different temperature dependency for $\mathbf{B}\parallel c$ and $\mathbf{B}\perp c$. We explain this by the different equienergy surface shape arising from the anisotropic and non-parabolic band structure, together with the energy-dependent electron-phonon scattering.