Dimers, webs, and local systems

For a planar bipartite graph $\mathcal G$ equipped with a $\mathrm{SL}_n$-local system, we show that the determinant of the associated Kasteleyn matrix counts "$n$-multiwebs" (generalizations of $n$-webs) in $\mathcal G$, weighted by their web-traces. We use this fact to study random $n$-multiwebs in graphs on some simple surfaces.