Let $K$ be a complete non-archimedean field and let $X$ be a $K$-analytic space in the sense of Berkovich of pure dimension $d$. Let $\varphi \colon X \to \mathbf{G}_m^r$ be a moment map to an analytic torus.
In [GJR21, Def. 3.11] the tropical skeleton of $\varphi \colon X \to \mathbf{G}_m^r$ is defined as $$S_\varphi(X) = \{ x \in X \mid d_\varphi(X,x) = d \}.$$ The quantity $d_\phi(X, x)$ is defined as follows: Given $x \in X$, there is an extension $\widetilde{\mathcal{H}}^\bullet(x) / \widetilde{K}^\bullet$ of graded fields in the sense of [Tem04]. Given a character $u \colon \mathbf{G}_m^r \to \mathbf{G}_m$, we get a function $\varphi^* u \colon X \to \mathbf{G}_m \hookrightarrow \mathbf{A}^1$ and we can consider $\widetilde{\chi}_x^\bullet(\varphi^* u) \in \widetilde{\mathcal{H}}^\bullet(x)$ (applying the character corresponding to $x$). Then $d_\phi(X, x)$ is defined to be the transcendence degree of the graded extension field of $\widetilde{K}^\bullet$ generated by all $\widetilde{\chi}_x^\bullet(\varphi^* u)$ [GJR21, Not. 3.6].
It is shown in [GJR21, Rem. 3.18] that $S_\varphi(X)$ can be written as a finite union of inverse images of the canonical skeleta of split tori so that it follows from [Duc12, Thm. 5.1] that $S_\varphi(X)$ carries a canonical structure as a piecewise linear space. In [GJR21, Rem. 4.12], weights are attached to the maximal faces of $S_\varphi(X)$.
So far I don't really have any intuition on how to think about these tropical skeletons. I only know that the tropicalization map induced by $\varphi$ maps $S_\varphi(X)$ finite-to-one onto the tropicalization of $X$ with respect to $\varphi$ and that the weights on the tropicalization can be described in terms of the weights on the skeleton. I think this implies that if the tropicalization is a smooth tropical variety (all weights are equal to one), then the tropicalization map is an isomorphism between the tropical skeleton and the tropicalization (at least on the full-dimensional part).
Are there any concrete enlightening examples of how the skeleton might look like? Is there an example where the piece-wise linear space $S_\varphi(X)$ cannot be embedded into a real affine space? Can this occur? I would also like to see an example where $S_\varphi(X)$ is not pure-dimensional if that may happen.
Also, at least when $\varphi$ is the analytification of a morphism of an algebraic variety to an algebraic torus, the tropicalization can be computed explicitely using initial degenerations as introduced by Speyer-Sturmfels. Can the skeleton also be computed (in the algebraic setting)?
[Duc12] – A. Ducros, Espaces de Berkovich, polytopes, squelettes et théorie des modèles.
[Tem04] – M. Temkin, On local properties of non-Archimedean analytic spaces. II.