Take the 2-minute tour ×  In Allermann-Rau '09, the authors
define the Chow groups of an arbitrary abstract tropical cycle.
In particular, one may take the tropical cycle to be the
tropicalization of a subvariety of a torus. They show that if
the ambient cycle is $\mathbb{R}^n$ (= the tropicalization of
the $n$-torus), then there is a well-defined intersection
product and we in fact have a Chow ring. A similar procedure is
used in Francois-Rau '10 to show that there is a Chow ring on
matroid varieties (= tropicalizations of linear subspaces of the
torus).  The standard notion of tropicalization transforms a
subvariety $X$ of a torus into a pure-dimensional polyhedral
complex $\text{Trop}(X)$ in Euclidean space. There is a notion
of "extended tropicalization" (see Payne '09, section 3) which
is applied to a toric variety $Y$. The result,
$\mathbf{Trop}(Y)$ is a partial compactification of the ordinary
tropicalization $\text{Trop}(Y \cap T)$, where $T$ is the dense
torus.  My question is, has anyone defined the