Lauren Williams pointed me toward http://front.math.ucdavis.edu/0609.5764 (the original reference for Postnikov's work) as well as http://front.math.ucdavis.edu/0706.2501 (an article by Postnikov, Speyer, and Williams with more of an emphasis on matchings and flows).  In this setting one has Plucker relations that are essentially the same thing as Kuo's graphical condensation.