  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.
