In real algebraic geometry, Stengle's Positivstellensatz can be used to characterize polynomials that are positive on a semialgebraic set.
Say that a tropical semialgebraic set is a subset of $\mathbb{R}^n$ defined by a finite sequence of tropical polynomial equations and tropical polynomial inequalities, or any finite union of these sets.
Given that so many theorems in algebraic geometry have a counterpart in tropical algebraic geometry, is there also a Positivstellensatz for characterizing tropical semialgebraic sets? Or something analog to Positivstellensatz in the tropical context?
If no, what are the main obstructions?
