  Let $(M,g)$ be a complete Riemannian $3$-manifold asymptotic to
Schwarzschild-anti-deSitter and with scalar curvature $R \geq - 6$. Building on
work of A.~Neves and G.~Tian and of the first-named author, we show that the
leaves of the canonical foliation of $(M, g)$ are the unique solutions of the
isoperimetric problem for their area. The assumption $R \geq -6$ is necessary.
  This is the first characterization result for large isoperimetric regions in
the asymptotically hyperbolic setting that does not assume exact rotational
symmetry at infinity.
