  The Ricci iteration is a discrete analogue of the Ricci flow. We give the
first study of the Ricci iteration on a class of Riemannian manifolds that are
not K\"ahler. The Ricci iteration in the non-K\"ahler setting exhibits new
phenomena. Among them is the existence of so-called ancient Ricci iterations.
As we show, these are closely related to ancient Ricci flows and provide the
first nontrivial examples of Riemannian metrics to which the Ricci operator can
be applied infinitely many times. In some of the cases we study, these ancient
Ricci iterations emerge (in the Gromov--Hausdorff topology) from a collapsed
Einstein metric and converge smoothly to a second Einstein metric. In the case
of compact homogeneous spaces with maximal isotropy, we prove a relative
compactness result that excludes collapsing.
