<p>We develop a conformal analog of the theory of Poisson bialgebras as well as a bialgebra theory of Poisson conformal algebras. We introduce the notion of Poisson conformal bialgebras, which are characterized by Manin triples of Poisson conformal algebras. A class of special Poisson conformal bialgebras called coboundary Poisson conformal bialgebras are constructed from skew-symmetric solutions of the Poisson conformal Yang-Baxter equation, whose operator forms are studied. Then we show that the semi-classical limits of conformal formal deformations of commutative and cocommutative antisymmetric infinitesimal conformal bialgebras are Poisson conformal bialgebras. Finally, we extend the correspondence between Poisson conformal algebras and Poisson-Gel’fand-Dorfman algebras to the context of bialgebras, that is, we introduce the notion of Poisson-Gel’fand-Dorfman bialgebras and show that Poisson-Gel’fand-Dorfman bialgebras correspond to a class of Poisson conformal bialgebras. Moreover, a construction of Poisson conformal bialgebras from pre-Poisson-Gel’fand-Dorfman algebras is given.</p>