We continue the study of multiple cluster structures in the rings of regular functions on
G
L
n
GL_n
,
S
L
n
SL_n
and
M
a
t
n
Mat_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
G
\mathcal {G}
corresponds to a cluster structure in
O
(
G
)
\mathcal {O}(\mathcal {G})
. Here we prove this conjecture for a large subset of Belavin–Drinfeld (BD) data of
A
n
A_n
type, which includes all the previously known examples. Namely, we subdivide all possible
A
n
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
SL_n
compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of
S
L
n
SL_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.