For GinUE matrices the (complex) Schur decomposition underpinned our ability to compute the eigenvalue PDF. This has a real analogue, although it is necessary to condition on the number of real eigenvalues. Consequently, the eigenvalue PDF is no longer absolutely continuous, but rather breaks up into sectors depending on the number of real eigenvalues. Nonetheless, as for real symmetric GOE matrices, structures giving rise to a Pfaffian point process result, the further development of which requires identifying particular skew-orthogonal polynomials. Questions relating to the distribution of real eigenvalues have a rich content, the development of which is facilitated by the identification of a simpler determinantal structure for the generating function, which permits the establishment of a local central limit theorem.