  In the recent publication [E. Kanzieper and G. Akemann, Phys. Rev. Lett. 95,
230201 (2005); arXiv: math-ph/0507058], an exact solution was reported for the
probability "p_{n,k}" to find exactly "k" real eigenvalues in the spectrum of
an "n" by "n" real asymmetric matrix drawn at random from Ginibre's Orthogonal
Ensemble (GinOE). In the present paper, we offer a detailed derivation of the
above result by concentrating on the proof of the Pfaffian integration theorem,
the key ingredient of our analysis of the statistics of real eigenvalues in the
GinOE. We also initiate a study of the correlations of complex eigenvalues and
derive a formula for the joint probability density function of all complex
eigenvalues of a GinOE matrix restricted to have exactly "k" real eigenvalues.
In the particular case of "k=0", all correlation functions of complex
eigenvalues are determined.
