  Real algebra is usually thought of as the study of certain kinds of preorders
on fields and rings. Among its core themes are the separation theorems known as
Positivstellens\"atze. However, there is a nascent subfield of real algebra
which studies preordered semirings and semifields, which is motivated by
applications to probability, graph theory and theoretical computer science,
among others. Here, we contribute to this subfield by developing a number of
foundational results for it, with two abstract Vergleichsstellens\"atze being
our main theorems. Our first Vergleichsstellensatz states that every semifield
preorder is the intersection of its total extensions. We apply this to derive
our second main result, a Vergleichsstellensatz for certain non-Archimedean
preordered semirings in which the homomorphisms to the tropical reals play an
important role. We show how this result recovers the existing
Vergleichsstellensatz of Strassen and (through the latter) the classical
Positivstellensatz of Krivine--Kadison--Dubois.
