The Gr\"obner stratification of a tropical variety

Each Gr\"obner stratum of a tropical variety is a connected set of points, all of which induce the same initial subscheme. The Gr\"obner stratification is a coarsening of the decomposition into Gr\"obner polyhedra, and has the advantage that it does not depend on a choice of compactification. We give an example of a curve over a field with non-trivial valuation whose Gr\"obner stratification is strictly finer than the coarsest polyhedral decomposition of the tropical variety. We also show that the Gr\"obner stratification of a locally matroidal tropical variety is completely determined by the underlying tropical variety.