For a finite set S of primes of a global field K and for sigma_1,...,sigma_e in Gal(K) we denote the field of totally S-adic numbers by K_tot,S; we denote the fixed field of sigma_1,...,sigma_e in K_tot,S by K_tot,S(sigma); and we denote the maximal Galois extension of K in K_tot,S(sigma) by K_tot,S[sigma]. We prove that for almost all sigma in Gal(K)^e, the absolute Galois group of K_tot,S[sigma] is isomorphic to the free product of \hat F_omega and a free product of local factors over S.