  In this paper, we study the interplay between tropical and analytic geometry
for closed subschemes of toric varieties. Let $K$ be a complete non-Archimedean
field, and let $X$ be a closed subscheme of a toric variety over $K$. We define
the tropical skeleton of $X$ as the subset of the associated Berkovich space
$X^{\rm an}$ which collects all Shilov boundary points in the fibers of the
Kajiwara--Payne tropicalization map. We develop polyhedral criteria for limit
points to belong to the tropical skeleton, and for the tropical skeleton to be
closed. We apply the limit point criteria to the question of continuity of the
canonical section of the tropicalization map on the multiplicity-one locus.
This map is known to be continuous on all torus orbits; we prove criteria for
continuity when crossing torus orbits. When $X$ is sch\"on and defined over a
discretely valued field, we show that the tropical skeleton coincides with a
skeleton of a strictly semistable pair, and is naturally isomorphic to the
parameterizing complex of Helm--Katz.
