Let $P_{1}\to B$ and $P_{2}\to B$ be smooth fiber bundles over the same base manifold $B$, and let us assume that the total spaces $(P_{1},\omega_{1})$ and $(P_{2},\omega_{2})$ are symplectic manifolds with corresponding symplectic forms $\omega_{1}$ and $\omega_{2}$. I am looking for examples of pairs $(P_{1},\omega_{1})$ and $(P_{2},\omega_{2})$ such that the fibered product
$P = P_{1}\times_{B} P_{2}$
is a known fiber bundle, namely, the total space $P$ is an explicitly known manifold.
Thanks.