  We study the tropicalizations of analytic subvarieties of normal toric
varieties over complete non-archimedean valuation fields. We show that a
Zariski closed analytic subvariety of a normal toric variety is algebraic if
its tropicalization is a finite union of polyhedra. Previously, the converse
direction was known by the theorem of Bieri and Groves. Over the field of
complex numbers, Madani, L. Nisse, and M. Nisse proved similar results for
analytic subvarieties of tori.
