  We construct symplectic groupoids integrating log-canonical Poisson
structures on cluster varieties of type $\mathcal{A}$ and $\mathcal{X}$ over
both the real and complex numbers. Extensions of these groupoids to the
completions of the cluster varieties where cluster variables are allowed to
vanish are also considered. In the real case, we construct
source-simply-connected groupoids for the cluster charts via the Poisson spray
technique of Crainic and M\u{a}rcu\c{t}. These groupoid charts and their
analogues for the symplectic double and blow-up groupoids are glued by lifting
the cluster mutations to groupoid comorphisms whose formulas are motivated by
the Hamiltonian perspective of cluster mutations introduced by Fock and
Goncharov.
