three piecewise-linear functions on a skeleton of ${X^{\rm an}}$ that were chosen to have the correct combinatorial properties and then tweaked so that we could lift those piecewise-linear functions to rational functions on $X$. The choice of these piecewise linear functions was inspired by Baker’s and Rabinoff’s construction [@BR Section 8]. Here, Baker and Rabinoff construct a faithful tropicalization for any curve in ambient dimension 3. Since they only consider faithful tropicalizations, they get to fix a skeleton beforehand (as opposed to a complete extended skeleton) and then construct an embedding that maps that skeleton isometrically onto its image (as opposed to out situation, where the completed extended skeleton depends on the embedding). This means that Baker and Rabinoff get much more freedom when picking their functions and only require a weaker lifting theorem.

Our main tool is a lifting theorem (Theorem \[lifting thm\]) of the second author [@J], that allows us to lift tropical meromorphic functions on a skeleton to the algebraic curve $X$. This theorem refines another lifting theorem of Baker and Rabinoff [@BR].

Similar questions to ours have been considered. For example in the works of Cartwright, Dudzik, Manjunath, and Yao [@CDMY] and Cheung, Fantini, Park, and 