  In tropical geometry, one studies algebraic curves using combinatorial
techniques via the tropicalization procedure. The tropicalization depends on a
map to an algebraic torus and the combinatorial methods are most useful when
the tropicalization has nice properties. We construct, for any Mumford curve
$X$, a map to a three-dimensional torus, such that the tropicalization is
isometric to a subgraph of the Berkovich space $X^{\rm an}$, called the
extended skeleton. In this case, we say the tropicalization is "fully
faithful."
  Additionally, given a map $X$ to a toric variety $Y$, which induces a fully
faithful tropicalization, we show that we can extend the map to $X \to Y \times
(\mathbf{P}^1)^n$ such that the new tropicalization is smooth and fully
faithful.
