I think that the easiest example of compact pseudo-Kähler manifold which does not admit any Kähler metric is the Kodaira-Thurston manifold. See for instance the introduction of
Yamada, Takumi, Ricci flatness of certain compact pseudo-Kähler solvmanifolds, J. Geom. Phys. 62, No. 5, 1338-1345 (2012). ZBL1239.53100.
