Profinite properties of residually free groups

Henry Wilton classified when a prime three-manifold $M$ has a residually free fundamental group $\pi_1 M$. We prove that the groups $\pi_1 M\times \mathbb Z^n$ are profinitely rigid within finitely generated residually free groups. We also establish other profinite invariants of the class of residually free groups such as coherence and subgroup separability. In the course of our proofs, we generalise a lemma of Wilton and Zalesskii on profinitely recognising when a central extension of groups splits.