  We prove the existence of multiple solutions to the Allen--Cahn--Hilliard
(ACH) vectorial equation (with two equations) involving a triple-well
(triphasic) potential with a small volume constraint on a closed parallelizable
Riemannian manifold. More precisely, we find a lower bound for the number of
solutions depending on some topological invariants of the underlying manifold.
The phase transition potential is considered to have a finite set of global
minima, where it also vanishes, and a subcritical growth at infinity. Our
strategy is to employ the Lusternik--Schnirelmann and infinite-dimensional
Morse theories for the vectorial energy functional. To this end, we exploit
that the associated ACH energy $\Gamma$-converges to the weighted
multi-perimeter for clusters, which combined with some deep theorems from
isoperimetric theory yields the suitable setup to apply the photography method.
Along the way, the lack of a closed analytic expression for the
multi-isoperimetric function for clusters imposes a delicate issue.
Furthermore, using a transversality theorem, we also show the genericity of the
set of metrics for which solutions to the ACH system are nondegenerate.
