  We reconstruct an explicit model of modified gravity in which a crossing of
the phantom divide can be realized. It is shown that the (finite-time) Big Rip
singularity appears in the model of modified gravity (i.e., in the so-called
Jordan frame), whereas that in the corresponding scalar field theory obtained
through the conformal transformation (i.e., in the so-called Einstein frame)
the singularity becomes the infinite-time one. Furthermore, we investigate the
relations between the scalar field theories with realizing a crossing of the
phantom divide and the corresponding modified gravitational theories by using
the inverse conformal transformation. It is demonstrated that the scalar field
theories describing the non-phantom phase (phantom one with the Big Rip) can be
represented as the theories of real (complex) $F(R)$ gravity through the
inverse (complex) conformal transformation. We also study a viable model of
modified gravity in which the transition from the de Sitter universe to the
phantom phase can occur. In addition, we explore the stability for the obtained
solutions of the crossing of the phantom divide under a quantum correction
coming from conformal anomaly.
