$\varphi$ is another metric graph with potentially infinite edges: the so-called completed extended skeleton $\Sigma = \Sigma({\varphi})$, which is a metric subgraph of ${X^{\rm an}}$. It was shown by Baker, Payne and Rabinoff [@BPR16] that $\operatorname{Trop}(X) = \operatorname{trop}_\varphi (\Sigma)$ and that $\operatorname{trop}_{\varphi}\!|_{\Sigma} \colon \Sigma \to \operatorname{Trop}(X)$ is a piecewise-linear, integral affine map of metric graphs. The tropicalization is called *fully faithful* if this map is an isometry. In particular, a fully faithful tropicalization admits a section $\operatorname{Trop}_{\varphi}(X) \to {X^{\rm an}}$. We can slightly relax those conditions: A tropicalization is called *totally faithful* if the map is an isometry when removing the vertices of $\Sigma$ that are infinitely far away.

We prove the following theorem (Theorem \[thm fully faithful\]) and a corollary (Theorem \[thm totally faithful\]) that is proved along the way.

\[ThmA\] Let $X$ be a smooth projective Mumford curve. Then there exist three rational functions $f_1, f_2, f_3$ on $X$ such that the tropicalization associated to the map $X \to (\mathbf{P}^1)^3, x \mapsto (f_1,f_2,f_3)$ is fully faithful.

Let $Y$ be a proper toric variety of dimension three. Then there exists a morphism $\varphi \colon X \to Y$ such that the induced tropicalization is totally faithful.

Our construction starts with 