  Statistical mechanics mappings provide key insights on quantum error
correction. However, existing mappings assume incoherent noise, thus ignoring
coherent errors due to, e.g., spurious gate rotations. We map the surface code
with coherent errors, taken as $X$- or $Z$-rotations (replacing bit or phase
flips), to a two-dimensional (2D) Ising model with complex couplings, and
further to a 2D Majorana scattering network. Our mappings reveal both
commonalities and qualitative differences in correcting coherent and incoherent
errors. For both, the error-correcting phase maps, as we explicitly show by
linking 2D networks to 1D fermions, to a $\mathbb{Z}_2$-nontrivial 2D
insulator. However, beyond a rotation angle $\phi_\text{th}$, instead of a
$\mathbb{Z}_2$-trivial insulator as for incoherent errors, coherent errors map
to a Majorana metal. This $\phi_\text{th}$ is the theoretically achievable
storage threshold. We numerically find $\phi_\text{th}\approx0.14\pi$. The
corresponding bit-flip rate $\sin^2(\phi_\text{th})\approx 0.18$ exceeds the
known incoherent threshold $p_\text{th}\approx0.11$.
