We introduce and study some mixed product Poisson structures on product manifolds
associated to Poisson Lie groups and Lie bialgebras. For quasitriangular Lie bialgebras,
our construction is equivalent to that of fusion products of quasi-Poisson G-manifolds
introduced by Alekseev, Kosmann- Schwarzbach, and Meinrenken. Our primary examples
include four series of holomorphic Poisson structures on products of flag varieties
and related spaces of complex semi-simple Lie groups.