Controlling formal fibers of principal prime ideals

Let (T, m) be a complete local (Noetherian) ring, S₀ the prime subring of T and p≠0 a regular and prime element of T. Given a finite set of incomparable prime ideals C = {Q₁,..., Q_n} of T such that either Q_i ∩ S₀ = (0) for all i or Q_i ∩ S₀ = pS₀ for all i, we provide necessary and sufficient conditions for T to be the completion of a local domain A such that p ε{lunate} A and the formal fiber of pA is semi-local with maximal ideals the elements of C. We also show that in a special case the domain A we construct is excellent.