We investigate Riemannian (non-Kaehler) Ricci flow solutions that develop finite-time Type-I singularities with the property that parabolic rescalings at the singularities converge to a Kaehler singularity model, specifically the "blowdown" shrinking soliton. Our results support the conjecture that the blowdown soliton is one of four dynamically stable singularity models in real dimension $n=4$. Our results also provide the first rigorous examples of non-Kaehler Ricci flow solutions that become asymptotically Kaehler in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kaehler metrics under the flow. This is joint work with James Isenberg and Natasa Sesum.