  Inhomogeneity in the ground state is an intriguing, emergent phenomenon in
magnetism. Recently, it has been observed in the magnetostructural channel of
the geometrically frustrated $\alpha$-NaMnO$_2$, for the first time in the
absence of active charge degrees of freedom. Here we report an in-depth
numerical and local-probe experimental study of the isostructural sister
compound CuMnO$_2$ that emphasizes and provides an explanation for the crucial
differences between the two systems. The experimentally verified, much more
homogeneous, ground state of the stoichiometric CuMnO$_2$ is attributed to the
reduced magnetoelastic competition between the counteracting magnetic-exchange
and elastic-energy contributions. The comparison of the two systems
additionally highlights the role of disorder and allows an understanding of the
puzzling phenomenon of phase separation in uniform antiferromagnets.
