We consider \(\mathbb {Z}_q\) -valued clock models on a regular tree, for general classes of ferromagnetic nearest neighbor interactions which have a discrete rotational symmetry. It has been proved recently that, at strong enough coupling, families of homogeneous Markov chain Gibbs states \(\mu _A\) coexist whose single-site marginals concentrate on \(A\subset \mathbb {Z}_q\) , and which are not convex combinations of each other [1]. In this note, we aim at a description of the extremal decomposition of \(\mu _A\) for \(|A|\ge 2\) into all extremal Gibbs measures, which may be spatially inhomogeneous. First, we show that in regimes of very strong coupling, \(\mu _A\) is not extremal. Moreover, \(\mu _A\) possesses a single-site reconstruction property which holds for spin values sent from the origin to infinity, when these initial values are chosen from A. As our main result, we show that \(\mu _A\) decomposes into uncountably many extremal inhomogeneous states. The proof is based on multi-site reconstruction which allows to derive concentration properties of branch overlaps. Our method is based on a new good site/bad site decomposition adapted to the A-localization property, together with a coarse graining argument in local state space.
