  In this paper we address two of the major foundational questions in the
theory of matroids over rings. First, we provide a cryptomorphic
axiomatisation, by introducing an analogue of the base polytope for matroids.
Second, we describe a parameter space for matroids over a valuation ring, which
turns out to be a tropical version of the Bott-Samelson varieties for the full
flag variety. Thus a matroid over a valuation ring is a sequence of flags of
tropical linear spaces a.k.a. valuated matroids.
