My attention was drawn this week to another part of Kamnitzer’s paper on MV-polytopes were he discusses clusters of primitive polytopes (observed by Anderson). Primitive polytopes are a finitie set (for the finite dimensional groups anyway) of polytopes that generate all MV-polytopes under Minkowski sum. MV-polytopes aren’t closed under Minkowski sum, but the primitive polytopes are grouped into clusters such that taking Minkowski sum within a cluster guarantees an MV-polytope. Prehaps I should emphasize, every MV-polytope can be written as the Minkowski sum of a set of primitive polytopes found in the same cluster.
The clusters correspond to tropical choices. When we have a tropical formula 
So the solution set to a tropcal formula is a union of cones. Each of these cones (generally overlapping) correspond to a tropical choice. If an MV polytope satisfies the tropical Plucker relations with a particular tropical choice — then it can be generated via Minkowski sum by polytopes in the cluster corresponding to that tropical choice.
I’m very interested in this because for $latex L\mathfrak{sl}_2$ I don’t have a satisfactory sense of tropical Plucker relations, but I do have a notion of “MV-polytopes” (not yet realized as moment map images of cycles, but functioning combinatorially in the same way) and of course Minkowski sum. So if I can observe primitives and clusters, I may gain some insight into what sort of tropical relations to expect.
