  We introduce the deformation space of pairs of distinct marked
$CP^1$-structures which have identical holonomy, so that a connected component
of this space is the quasi-Fuchsian space. By the uniformization theorem of
Riemann surfaces, every connected component of the space homomorphically maps
to the product of associated Teichm\"uller spaces (minus its diagonal, when
appropriate).
  In this paper, we show that this uniformization map is a complete local
branched covering map. This is a partial generalization of Bers' simultaneous
uniformization theorem. As a corollary, indeed, we reprove the simultaneous
uniformization theorem without the measurable Riemann mapping theorem.
