The exchange relation in a cluster algebra is
\begin{align}
x\_k' = \frac{1}{x\_k} (\prod\_{j \to k}x\_j + \prod\_{k \to j} x\_j).
\end{align}
Do we have some tropical version of this relation? Are there some references? Thank you very much.



Tropical version of exchange relations in cluster algebras
This maybe too naive and not helpful to your problem: pretend $x\_i$ are positive real number, and let $x\_i = e^{-t a\_i}$ for $a\_i \in \mathbb{R}$. One can consider the limit $t \to \infty$ and taking $log$ on both sides to get
$$ a\_k' = \min(\sum\_{j \to k} a\_j, \sum\_{k \to l} a\_l) - a\_k $$
Of course, this is the same thing as doing the tropical sum (the usual min) and tropical product/quotient (the usual $+/-$). 


The paper by Gross-Hacking-Keel-Kontsevich, [Canonical bases for cluster algebras](https://arxiv.org/abs/1411.1394), maybe is more relevant. 


