  Given a consistent bipartite graph $\Gamma$ in $T^2$ with a complex-valued
edge weighting $\mathcal{E}$ we show the following two constructions are the
same. The first is to form the Kasteleyn operator of $(\Gamma, \mathcal{E})$
and pass to its spectral transform, a coherent sheaf supported on a spectral
curve in $(\mathbb{C}^\times)^2$. The second is to form the conjugate
Lagrangian $L \subset T^* T^2$ of $\Gamma$, equip it with a brane structure
prescribed by $\mathcal{E}$, and pass to its mirror coherent sheaf. This lives
on a stacky toric compactification of $(\mathbb{C}^\times)^2$ determined by the
Legendrian link which lifts the zig-zag paths of $\Gamma$ (and to which the
noncompact Lagrangian $L$ is asymptotic). We work in the setting of the
coherent-constructible correspondence, a sheaf-theoretic model of toric mirror
symmetry. We also show that tensoring with line bundles on the compactification
is mirror to certain Legendrian autoisotopies of the asymptotic boundary of
$L$.
