Looking for an example of a symplectic manifold $(M,\omega)$ that is not symplectomorphic to $(M,-\omega)$.

In particular this means that $M$ must be chiral (i.e. doesn't admit an orientation-reversing diffeomorphism).

For a topological obstruction, I think it would be enough to find $(M,\omega)$ such that $Diff(M)$ acts trivially on $H^2(M)\neq0$.

A complex projective variety defined by real equations won't work, because the complex conjugation map is antisymplectic.