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The correspondence between on-shell diagrams in maximally supersymmetricYang-Mills theory and cluster varieties in the Grassmannian remains largelyunexplored beyond the planar limit. In this article, we describe a systematicprogram to survey such 'on-shell varieties', and use this to provide a completeclassification in the case of $G(3,6)$. In particular, we find exactly 24top-dimensional varieties and 10 co-dimension one varieties in $G(3,6)$---up toparity and relabeling of the external legs. We use this case to illustrate someof the novelties found for non-planar varieties relative to the case ofpositroids, and describe some of the features that we expect to hold moregenerally.

Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams