The study on the large-time behaviour of solutions to the 3D incompressible anisotropic Navier–Stokes (ANS) equations is very recent. Powerful tools designed for the Navier–Stokes equations with full Laplacian dissipation such as the Fourier splitting method no longer apply to the case when there is only horizontal dissipation. For the whole space R 3 , as t → ∞ , solutions of the ANS equations converge to the trivial solution and the convergence rate is algebraic. This paper is devoted to the case when the spatial domain Ω is T 2 × R . Our results reveal that the large-time behaviour for T 2 × R is quite different from that for R 3 . We show that any small initial velocity field u 0 ∈ H 2 ( Ω ) leads to a unique global solution u that remains small in H 2 ( Ω ) . More importantly, as t → ∞ , the velocity field u converges to a nontrivial steady state. The first two components of the steady state are given by the horizontal average of the first two components of u 0 while the third component vanishes. In addition, this convergence is exponentially fast.