  In this paper, we will introduce the notion of {\it conjugate phase
retrieval}, which is a relaxed definition of phase retrieval allowing recovery
of signals up to conjugacy as well as a global phase factor. It is known that
frames of real vectors are never phase retrievable on ${\mathbb C}^M$ in the
ordinary sense, but we show that they can be conjugate phase retrievable in
complex vector spaces. We continue to develop the theory on conjugate phase
retrievable real frames. In particular, a complete characterization of
conjugate phase retrievable real frames on ${\mathbb C}^2$ and ${\mathbb C}^3$
is given. Furthermore, we show that a generic real frame with at least $4M - 6$
measurements is conjugate phase retrievable in ${\mathbb C}^M$ for $ M \ge 4.$
