faithful tropicalizations is open for non-Mumford curves.

Construction of fully faithful tropicalization in 3-space {#sec:construction}
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In this section, $X$ will denote a Mumford curve over a complete, algebraically closed, non-Archimedean valued field $K$ with analytification ${X^{\rm an}}$ and skeleton $\Gamma$. We take $G$ to be a graph model of $\Gamma$ with vertex set $V = V(G)$ and edge set $E = E(G)$.

After possibly subdividing, we assume that $G$ has edges $e_1,\dots,e_g$ that form the complement of a spanning tree, $T \subseteq E$, and that no two edges $e_i, e_j$ share a vertex.

We will define three piecewise-linear functions $F_1, F_2, F_3$ on $\Gamma$ whose graphs are depicted in Figures \[fig:F1-graph\] to \[fig:F2-graphT\]. To construct these piecewise-linear functions, we consider divisors on $\Gamma$ and use the following lifting theorem.

\[lifting thm\] Let $D$ be a divisor on $X$ of degree $g$. Given any break divisor $B = x_1 + \dots + x_g$ on $\Gamma$ supported on 2-valent points, if $\tau_*D - B$ is principal then there exist liftings $x_1',\dots,x_g' \in X(K)$ such that $\tau_*x_i' = x_i$ and such that $D - \sum_{i = 1}^g x_i'$ is a principal divisor.

Theorem 3.2 of [@J].

Another equivalent way of writing this 