  On $4$-symmetric symplectic spaces, invariant almost complex structures -- up
to sign -- arise in pairs. We exhibit some $4$-symmetric symplectic spaces,
with a pair of "natural" compatible (usually not positive) invariant almost
complex structures, one of them being integrable and the other one being
maximally non integrable (i.e. the image of its Nijenhuis tensor at any point
is the whole tangent space at that point). The integrable one defines a
pseudo-K\"ahler Einstein metric on the manifold, and the non integrable one is
Ricci Hermitian (in the sense that the almost complex structure preserves the
Ricci tensor of the associated Levi Civita connection) and special in the sense
that the associated Chern Ricci form is proportional to the symplectic form.
