  In this paper we provide a classification of all Moishezon twistor spaces on
the connected sum of four complex projective planes. This is given by means of
the anticanonical system of the twistor spaces. In particular, we show that the
anticanonical map is birational, two to one over the image, or otherwise the
image of the anticanonical map is a rational surface. We also obtain the
structure of the images of the anticanonical map in each of the three cases in
explicit forms.
