Let $P^2 \tilde \times \mathbb R^2$ and $K^2 \tilde \times \mathbb R^2$ be the twisted plane bundle over the real projective plane and the Klein bottle so that their double covers are $S^2 \times \mathbb R^2$ and $T^2 \times \mathbb R^2$, respectively.

Can we find a Kahler surface $M$ so that $M$ is homeomorphic to $P^2 \tilde \times \mathbb R^2$ or $K^2 \tilde \times \mathbb R^2$?