  We analyze the elementary theory of certain fields of totally S-adic
algebraic numbers that were introduced and studied by Geyer-Jarden and
Haran-Jarden-Pop. In particular, we provide an axiomatization of these theories
and prove their decidability, thereby giving a common generalization of
classical decidability results of Jarden-Kiehne, Fried-Haran-V\"olklein and
Ershov.
