Congruences on tropical rational function semifields and tropical curves

We define tropical rational function semifields $\overline{\mathit{T}(X_1,\dots ,X_n)}$ and prove that a tropical curve $\Gamma$ is realized (except for points at infinity) as the congruence variety $V\subset {\mathit{R}}^n$ associated with a congruence on $\overline{\mathit{T}(X_1,\dots ,X_n)}$ by giving a specific map $\Gamma \to V$. Also, we shed light on the relation between congruences $E$ on $\overline{\mathit{T}(X_1,\dots ,X_n)}$ and congruence varieties associated with them and reveal the quotient semifield $\overline{\mathit{T}(X_1,\dots ,X_n)}\mathrm{/}E$ to play the role of coordinate rings that determine isomorphism classes of affine varieties in the classical algebraic geometry.