In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of the usual ring-theoretic scheme theory to tropical algebras, but rather a delicately chosen modification (see, especially, his definition of rational functions and his method of handling localisations).
On the other hand, Durov's thesis (available here) constructs a very general framework for doing algebraic geometry, and in particular, defining schemes, over things more general than commutative rings. This setting happens to include semirings such as the tropical semifield. So Durov's work gives us a theory of tropical schemes as well. (Actually, from what I can tell, the resulting theory depends on some choice of localisation theory.)
Has anyone out there taken the time to compare these two theories of tropical schemes and understand how they are related?