  We complete the program of Ref. [3,5] of connecting totally non-negative
Grassmannians to the reality problem in KP finite-gap theory via the assignment
of real regular divisors on rational degenerations of M-curves for the class of
real regular multi-line soliton solutions of KP II equation whose asymptotic
behavior has been combinatorially characterized by Chakravarthy, Kodama and
Williams. We use the plabic networks in the disk introduced by Postnikov to
parametrize positroid cells in totally nonnegative Grassmannians. The boundary
of the disk corresponds to the rational curve associated to the soliton data in
the direct spectral problem, and the bicolored graph is the dual of a reducible
curve G which is the rational degeneration of a regular M-curve whose genus g
equals the number of faces of the network diminished by one. We assign systems
of edge vectors to the planar bicolored networks. The system of relations
satisfied by them has maximal rank and may be reformulated in the form of edge
signatures.. We prove that the components of the edge vectors are rational in
the edge weights with subtraction free denominators and provide their explicit
expressions in terms of conservative and edge flows. The edge vectors rule the
value of the KP wave function at the double points of G, whereas the signatures
at the vertices rule the position of the divisor in the ovals. We prove that
the divisor satisfies the conditions of Dubrovin-Natanzon for real finite-gap
solutions: there is exactly one divisor point in each oval except for the one
containing the essential singularity of the wave function. The divisor may be
explicitly computed using the linear relations at the vertices of the network.
We explain the role of moves and reductions in the transformation of both the
curve and the divisor for given soliton data, and apply our construction to
some examples.
