It is known [M4] that Kℂ-orbits S and Gℝ-orbits S' on a complex flag manifold are in one-to-one correspondence by the condition that S ∩ S' is nonempty and compact. It is possible to replace Kℂ by some conjugate xKℂx−1 so that the correspondence is preserved. We investigate the sets C(S) of such x for various orbits S and their relations with each other. We prove that for classical groups the intersection C = ∩S C(S) equals D0Z where D0 = D0/Kℂ is the universal domain in Gℂ/Kℂ introduced in [AG] and Z is the center of G. As a corollary we prove that D0 is Stein for classical groups. Moreover we conjecture that C(S)0 = D0 for generic S where C(S)0 is the connected component of C(S) containing the identity.