For a variety X (over some proper fields), if Trop(X) is a tropicalization of X, then 
 we know that Trop(X) is a polyhedral complex. If we consider the interior of the support of that polyhedral complex, then does it have some geometric meaning (in the sense of tropicalization) ?



Boundary of a tropical variety.
First of all, $\mathrm{Trop}(X)$ is a set which can be equipped with the structure of a polyhedral complex; there is not a canonical way to do it. However, there are better and worse choices. In particular, one can choose the polyhedral structure so that, if $w$ and $w'$ are in the relative interiors of the same face, then the initial ideals $\mathrm{In}\_w(X)$ and $\mathrm{In}\_{w'}(X)$ are equal. 


Let $X$ be pure of dimension $d$ and choose such a polyhedral structure. For such a choice, if $w$ is in the relative interior of a face then the automorphisms of $\mathrm{In}\_w(X)$ contain a $d$-dimensional torus. I wrote this up in Section 2.2 of [my thesis](http://www.math.lsa.umich.edu/~speyer/thesis.pdf) (apologies for the horrible writing). You can also extract this from the discussion in Section 9.2 of Sturmfels [Solving Systems of Polynomial Equations](http://math.berkeley.edu/~bernd/cbms.pdf). Remark 3.4.4 in Sturmfels and Maclagan's [textbook-in-progress](http://homepages.warwick.ac.uk/staff/D.Maclagan/papers/TropicalBook23.8.13.pdf) states this result as well, and the surrounding discussion is probably the clearest of any of these sources.


