Existentially Incomplete Tame Models and a Conjecture of Ellentuck

We construct a recursive ultrapower F/U such that F/U is a tame 1‐model in the sense of [6, §3] and FU is existentially incomplete in the models of II2 arithmetic. This enables us to answer in the negative a question about closure with respect to recursive fibers of certain special semirings Γ of isols termed tame models by Barback. Erik Ellentuck had conjuctured that all such semirings enjoy the closure property in question. Our result is that while many do, some do not.