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A symplectic realization of a Poisson manifold M is a symplectic manifold Σ together with a Poisson map from Σ to M which is a surjective submersion. If this map has a cross section with the Lagrangian image Λ, then a neighborhood of Λ in Σ can be canonically equipped with the structure of a local symplectic groupoid for which Λ is the unit space and M is the object space. The space A of smooth functions on the formal neighborhood of Λ in Σ is a Poisson algebra. The groupoid operations induce dual mappings between the Poisson algebra A and the Poisson algebra of smooth functions on M. We give a self-contained algebraic description of the Poisson algebra A and use it to define a formal symplectic groupoid over the Poisson manifold M. We show that to each (natural) formal deformation quantization on M one can relate a formal symplectic groupoid over M. Finally, we construct a unique formal symplectic groupoid "with separation of variables" over an arbitrary Kähler-Poisson manifold. |