Finitely generated congruences on tropical rational function semifields

We prove that the congruence on the tropical rational function semifield in $n$-variables associated with a subset $V$ of ${\mathit{R}}^n$ is finitely generated if and only if the closure of $V$ is a finite union of $\mathit{R}$-rational polyhedral sets. With this fact, we characterize rational function semifields of tropical curves.