We show that the morphology of the initial monolayers of InP on Al0.48In0.52As grown by metalorganic vapor-phase epitaxy does not follow the expected layer-by-layer growth mode of lattice-matched systems, but instead develops a number of low-dimensional structures, e.g., quantum dots and wires. We discuss how the macroscopically strain-free heteroepitaxy might be strongly affected by local phase separation/alloying-induced strain and that the preferred aggregation of adatom species on the substrate surface and reduced wettability of InP on AlInAs surfaces might be the cause of the unusual (step) organization and morphology.