  The tropicalization of an algebraic variety X is a combinatorial shadow of X,
which is sensitive to a closed embedding of X into a toric variety. Given a
good embedding, the tropicalization can provide a lot of information about X.
We construct two types of these good embeddings for Mumford curves: Fully
faithful tropicalizations, which are embeddings such that the tropicalization
admits a section to the associated Berkovich space $X^{an}$ of X, and smooth
tropicalizations. We also show that a smooth curve that admits a smooth
tropicalization is necessarily a Mumford curve. Our key tool is a variant of a
lifting theorem for rational functions on metric graphs.
