In this paper Vigeland shows how one can define a group law on subset of a tropical elliptic curve, so that this group is homeomorphic to $S^1$. It is not clear to me what is the relationship between the definition of tropical elliptic curve in the paper to tropicalization of (a toric embedding of a dense open subset of) ellptic curve.
Therefore my question.
Let $A$ be an Abelian variety over a non-Archimedean varietyfield $K$, and suppose that value group of $K$ is $\mathbb R$. Is it possible to find an open set $U \subset A$ and an affine embedding $U \subset \mathbb{G}_m^n$ such that the image of $U$ in $\mathbb{R}^n$ under the valuation map can be endowed with a group structure (after restricting to some open subset of this image). If yes, is there a relationship between this group structre and the group structure of the Abelian variety $A$?