We present results on the non-linear dynamics of inhomogeneous cosmological models with irrotational dust and a positive cosmological constant, considering, in particular, a wide class with vanishing magnetic Weyl tensor. For those patches of the universe that do not recollapse to a singularity we find a unique attractor, representing a de Sitter vacuum phase. For the (re-)collapsing regions we find a family of (Kasner) attractors, so that generically these regions fall in spindle-like

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... ties. These results give substantial support to the idea that the universe can be very inhomogeneous on ultra-large, super-horizon scales, whit observers living in those (almost) isotropic regions that emerge from an inflationary phase.