it seems that Palais meant something else, or made a mistake

the map which switches two factors is symplectic, and we need antisymplectic

I would conjecture that it is always $CP^1 \times T^2$, because I see no other examples

Understood. Now simplicity is never assumed in the definition of stability (see Lubcke-Teleman). Sorry for a confusion.

However, there are examples of non-formal (hence, non-Kahler) symplectic manifolds satisfying hard Lefschetz; see arxiv.org/abs/math/0403067, "The Lefschetz property, formality and blowing up in symplectic geometry", by Gil R. Cavalcanti

non-projective Kahler 3-manifolds are in fact rare: they all admit a holomorphic 2-form; its radical gives a 1-dimensional foliation on a manifold. Its curvature by Brunella is represented by a positive current, unless each leaf of this foliation has closure isomorphic to a rational line. In the second case we already have a good idea about the Mori fibration. In the first case, the canonical bundle is pseudoeffective, which is not far from K_x being nef.