  We constructed in a previous work the $\Phi^4_3$ measures on compact
boundaryless $3$-dimensional Riemannian manifolds as some invariant probability
measures of some Markovian dynamics. We prove in the present work that these
dynamics have unique invariant probability measures. This is done by using an
explicit coupling by change of measure that does not require any a priori
information on the support of the law of the solution to the dynamics. The
coupling can be used to see that the semigroup generated by the dynamics
satisfies a Harnack-type inequality, which entails that the semigroup has the
strong Feller property.
