Star products with separation of variables admitting a smooth extension

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.