  We classify, up to a local isometry, all non-Kahler almost Kahler 4-manifolds
for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose
Ricci tensor is invariant with respect to the almost complex structure.
Equivalently, such almost Kahler 4-manifolds satisfy the third curvature
condition of A. Gray. We use our local classification to show that, in the
compact case, the third curvature condition of Gray is equivalent to the
integrability of the corresponding almost complex structure.
