In Section \[sec:injective\] we show that if our parameters meet the conditions stated in Section \[sec:params\], the map induced by the lifts of the functions from Section \[sec:construction\] induces a totally faithful tropicalization.

In Section \[sec:compactifications\] we complete the proof of Theorem \[ThmA\] by showing that the conditions in Section \[sec:params\] can always be met and we show that tropicalizations is indeed already fully faithful.

In Section \[sec:res-of-singularities\] we prove Theorem \[ThmB\] via a resolution procedure for embedded tropical curves.

In Section \[sec:genus2\] we exhibit our lifting techniques on a more specific example of a genus $2$ skeleton.

Acknowledgements
----------------

The authors would like to thank Joe Rabinoff for fruitful discussions and Matt Baker and Walter Gubler for comments on a preliminary draft.

Preliminaries {#sec:prelim}
=============

Throughout this paper, $K$ will denote an algebraically closed field which is complete with respect to a non-trivial, non-Archimedean absolute value $|\cdot|_K$. We denote the value group of $K$ by $\Lambda \coloneqq \log |K^\times| \subseteq {\mathbf}R$.

Tropicalization of curves in P\^n {#sec:classical-tropicalization}
---------------------------------

Recall that *tropical projective space* is defined by $${\mathbf}{TP}^n = \Big(({\mathbf}R \cup \{-\infty\})^{n+1} \setminus \{(-\infty,\dots,-\infty)\}\Big)/ {\mathbf}R.$$ The action of ${\mathbf}R$ is by “tropical scalar multiplication”: $$\lambda \cdot (a_0,\dots,a_n) = (a_0 + 