  The correspondence between on-shell diagrams in maximally supersymmetric
Yang-Mills theory and cluster varieties in the Grassmannian remains largely
unexplored beyond the planar limit. In this article, we describe a systematic
program to survey such 'on-shell varieties', and use this to provide a complete
classification in the case of $G(3,6)$. In particular, we find exactly 24
top-dimensional varieties and 10 co-dimension one varieties in $G(3,6)$---up to
parity and relabeling of the external legs. We use this case to illustrate some
of the novelties found for non-planar varieties relative to the case of
positroids, and describe some of the features that we expect to hold more
generally.
