  We prove that a semiring isomorphism between the rational function semifields
of two tropical curves induces an expansive map between those tropical curves.
This semiring isomorphism and the expansive map respect zeros and poles of
rational functions with their degrees. As a corollary, we show that the
automorphism group of a tropical curve is isomorphic to the
$\boldsymbol{T}$-algebra automorphism group of its rational function semifield,
where $\boldsymbol{T} := (\boldsymbol{R} \cup \{ -\infty \},
\operatorname{max}, +)$ is the tropical semifield. Finally, we describe all
semiring automorphisms of rational function semifields of all tropical curves.
