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 Z2-nontrivial 2D insulator. However, beyond a rotation angle ϕth, instead of a Z2-trivial insulator as for incoherent errors, coherent errors map to a Majorana metal. This ϕth is the theoretically achievable storage threshold. We numerically find ϕth≈0.14π. The corresponding bit-flip rate sin2(ϕth)≈0.18 exceeds the known incoherent threshold pth≈0.11.