  We investigate a corner of the Bianchi models that has not received much
attention: "extended FLRW models" (eFLRW) defined as a cosmological model with
underlying anisotropic Bianchi geometry that nevertheless expands isotropically
and can be mapped onto a reference FLRW model with the same expansion history.
In order to investigate the stability and naturalness of such models in a
dynamical systems context, we consider spatially homogeneous models that
contain a massless scalar field $\varphi$ and a non-tilted perfect fluid
obeying an equation of state $p=w\rho$. Remarkably, we find that matter
anisotropies and geometrical anisotropies tend to cancel out dynamically.
Hence, the expansion is asymptotically isotropic under rather general
conditions. Although extended FLRW models require a special matter sector with
anisotropies that are 'fine-tuned" relative to geometrical anisotropies, our
analysis shows that such solutions are dynamically preferred attractors in
general relativity. Specifically, we prove that all locally rotationally
symmetric Bianchi type III universes with space-like $\nabla_\mu\varphi$ are
asymptotically shear-free, for all $w\in[-1,1]$. Moreover, all shear-free
equilibrium sets with anisotropic spatial curvature are proved to be stable
with respect to all homogeneous perturbations for $w\geq -1/3$.
