  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.
