There exists a deep correspondence between a class of physically important functions --called ``on-shell functions''-- and certain (cluster variety) subspaces of Grassmannian manifolds, endowed with a volume form that is left invariant under cluster coordinate transformations. These are called ``on-shell varieties'' (which may or may not include all cluster varieties). It is easy to prove that the number of on-shell varieties is finite, from which it follows that the same is true for on-shell functions. This is powerful and surprising for physics, because these on-shell functions encode complete information about perturbative quantum field theory. \par In this talk, I describe the details of this correspondence and how it is constructed and give the broad physics motivations for obtaining a more systematic understanding of on-shell cluster varieties. I outline a general, brute-force strategy for classifying these spaces; and describe the results found by applying this strategy to the case of Gr(3,6).