  In this paper, we first introduce the definition of a Hom-Poisson bialgebra
and give an equivalent descriptions via the Manin triple of Hom-Poisson
algebras. Also we introduce notions of $\mathcal{O}$-operator on a Hom-Poisson
algebra, post-Hom-Poisson algebra and quasitriangular Hom-Poisson bialgebra,
and present a method to construct post-Hom-Poisson algebras. Finally, we show
that a quasitriangular Hom-Poisson bialgebra naturally yields a
post-Hom-Poisson algebra.
