Tropical geometry studies the semiring $(\mathbb R, \min,+)$. I'm wondering if anything is lost by studying semirings over some field other than $\mathbb R$?
It's obvious that finite fields and $\mathbb C$ won't work because $\min$ isn't well defined in these fields. But what about subfields of $\mathbb R$ like the rational or computable numbers? Is anything from the standard theory lost in this substitution?