  Motivated by the problem of giving a bijective proof of the fact that the
birational RSK correspondence satisfies the octahedron recurrence, we define
interlacing networks, which are certain planar directed networks with a rigid
structure of sources and sinks. We describe an involution that swaps paths in
these networks and leads to Pl\"{u}cker-like three-term relations among path
weights. We show that indeed these relations follow from the Pl\"{u}cker
relations in the Grassmannian together with some simple rank properties of the
matrices corresponding to our interlacing networks. The space of matrices
obeying these rank properties forms the closure of a cell in the matroid
stratification of the totally nonnegative Grassmannian. Not only does the
octahedron recurrence for RSK follow immediately from the three-term relations
for interlacing networks, but also these relations imply some interesting
identities of Schur functions reminiscent of those obtained by Fulmek and
Kleber. These Schur function identities lead to some results on Schur
positivity for expressions of the form $s_{\nu}s_{\rho} - s_{\lambda}s_{\mu}$.
