  Given a complex manifold $M$ with an open dense subset $\Omega$ endowed with
a pseudo-Kaehler form $\omega$ which cannot be smoothly extended to a larger
open subset, we consider various examples where the corresponding
Kaehler-Poisson structure and a star product with separation of variables on
$(\Omega, \omega)$ admit smooth extensions to $M$. We suggest a simple
criterion of the existence of a smooth extension of a star product and apply it
to these examples.
