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We continue the study of multiple cluster structures in the rings of regular functions on $G{L}_n$, $S{L}_n$and $Ma{t}_n$that are compatible with Poisson–Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on a semisimple complex group $\mathcal{G}$corresponds to a cluster structure in $\mathcal{O}\left(\mathcal{G}\right)$. Here we prove this conjecture for a large subset of Belavin–Drinfeld (BD) data of $A_n$type, which includes all the previously known examples. Namely, we subdivide all possible $A_n$type BD data into oriented and non-oriented kinds. We further single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any oriented BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson–Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on $S{L}_n$compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of $S{L}_n$equipped with two different Poisson-Lie brackets. Similar results hold for aperiodic non-oriented BD data, but the analysis of the corresponding regular cluster structure is more involved and not given here. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address these situations in future publications.