What does the Riemann-Roch theorem say in the tropical world?

Background: One way to approach this would be to build up the machinery of line bundles (or maybe coherent sheaves?). A different, more immediately geometric approach might be called the ``Brill-Noether'' approach. This requires only two ingredients:

1. ``plane curve with ordinary nodes'' and
2. If $A$, $B$ and $C$ are curves with a common point of intersection, what is `` $A\cap B - A\cap C$'', the ``residual intersection of $C$ in $A\cap B$''. If $\mathcal{O}_x$ is the local ring of $x$ on $A$, and $f_B$, $f_C$ are the images in $\mathcal{O}_x$ of the equations of $B$ and $C$, then in the classical case the parts of the intersections $A\cap B$ and $A\cap C$ supported at $x$ are represented by the ideals $(f_B)$ and $(f_C)$ in $\mathcal{O}_x$, and the residual is represented by the ideal
   
   $$(f_B:f_C) := \{g\in \mathcal{O}_x \, \vert \, gf_C \in (f_B)\} \, .$$
   
   
   The early work on Riemann-Roch treated only the case where $A$ is smooth at $x$. Then $A\cap B$ at $x$ is represented just by a multiplicity, and residuation is just subtraction. When $A$ is arbitrary, things still work because $\mathcal{O}_x$ is a Gorenstein ring for any smooth curve, no matter how singular.

Question: How do these notions play out for tropical plane curves?

(contributed by David Eisenbud)

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