By definition a semiring is zero-sum free if $x\oplus y=\bar{0}$ implies $x=y=\bar{0}$ for all $x,y \in \mathbb{K}$. The tropical semiring is zero-sum free. I'm using this definition for the tropical semiring: $(\mathbb{K}, \oplus, \otimes, \bar{0}, \bar{1})=(\mathbb{R}\cup \{-\infty, +\infty\}, \min, +, +\infty, 0)$ as in Table 1 in this paper by Mohri.
However, for the tropical semiring, is it not possible that either $x=\bar{0}$ or $y=\bar{0}$, since $x\oplus y = \min(x,y)$ (with the other from $\mathbb{R}^+$)?
I.e. must it be the case that both $x=\bar{0}$ and $y=\bar{0}$? Or is there some subtlety when the semiring is defined over all $\mathbb{R}$?
(Related note, I saw in this paper, Remark 2.4, that "An idempotent semiring is necessary zero-sum free.". I can see that that's true if it's necessary that $x=y=\bar{0}$, since we need $\bar{0}\oplus\bar{0}=\bar{0}$, but otherwise it's not clear to me).
For the tropical semiring in your definition, $\bar 0 = +\infty$. Thus, $min(x,y)=\bar 0=+\infty$ actually means $x=y=\infty$.
(And if your semiring is idempotent and you have x+y=0 then the following is true: $$0=x+y=(x+x)+y=x+(x+y)=x+0=x$$ It follows $x=0$. Analogously for y.)