  In this paper, we investigate tropical secant varieties of ordinary linear
spaces. These correspond to the log-limit sets of ordinary toric varieties; we
show that their interesting parts are combinatorially isomorphic to a certain
natural subcomplex of the complex of regular subdivisions of a corresponding
point set, and we display the range of behavior of this object. We also use
this characterization to reformulate the question of determining Barvinok rank
into a question regarding regular subdivisions of products of simplices.
