  Tropical geometry gives a bound on the ranks of divisors on curves in terms
of the combinatorics of the dual graph of a degeneration. We show that for a
family of examples, curves realizing this bound might only exist over certain
characteristics or over certain fields of definition. Our examples also apply
to the theory of metrized complexes and weighted graphs. These examples arise
by relating the lifting problem to matroid realizability. We also give a proof
of Mn\"ev universality with explicit bounds on the size of the matroid, which
may be of independent interest.
