  Maximal minors of Kasteleyn sign matrices on planar bipartite graphs in the
disk count dimer configurations with prescribed boundary conditions, and the
weighted version of such matrices provides a natural parametrization of the
totally non-negative part of real Grassmannians (see Ref. [53,42,43,57,7]). In
this paper we provide a geometric interpretation of Kasteleyn theorem showing
that the geometric signatures introduced in Ref. [5] are Kasteleyn. We apply
this geometric characterization to explicitly solve the associated system of
relations and give an alternative proof to that in Ref. [57] that the
parametrization of positroid cells induced by Kasteleyn weighted matrices
coincides with that of Postnikov boundary measurement map. Finally we use
Kasteleyn system of relations to construct real regular KP divisors on rational
degeneration of $\mathtt M$--curves considerably simplifying the approach to
the same question in Ref. [6] for the given class of graphs.
