  The magnetocrystalline anisotropy energy of atomically ordered
$\mathrm{L}1_0$ FeNi (the meteoritic mineral tetrataenite) is studied within a
first-principles electronic structure framework. Two compositions are examined:
equiatomic Fe$_{0.5}$Ni$_{0.5}$ and an Fe-rich composition,
Fe$_{0.56}$Ni$_{0.44}$. It is confirmed that, for the single crystals modelled
in this work, the leading-order anisotropy coefficient $K_1$ dominates the
higher-order coefficients $K_2$ and $K_3$. To enable comparison with
experiment, the effects of both imperfect atomic long-range order and finite
temperature are included. While our computational results initially appear to
undershoot the measured experimental values for this system, careful scrutiny
of the original analysis due to N\'{e}el et al. [J. Appl. Phys. 35, 873 (1964)]
suggests that our computed value of $K_1$ is, in fact, consistent with
experimental values, and that the noted discrepancy has its origins in the
nanoscale polycrystalline, multivariant nature of experimental samples, that
yields much larger values of $K_2$ and $K_3$ than expected a priori. These
results provide fresh insight into the existing discrepancies in the literature
regarding the value of tetrataenite's uniaxial magnetocrystalline anisotropy in
both natural and synthetic samples.
