The fundamental QFT is formulated in a separable Hilbert space. But mostly approaches in condensed matter physics, e. g. thermal field dynamics, use a non-separable Hilbert space. It looks like it is not easy to find how a theory with separable Hilbert space can act like a basement for non-separable Hilbert space of condensed matter physics. Is it really (open) problem or not? if yes, whether it is possible to hope that the decoherence theory can shed light?
That is the same situation as in QFT, as soon as you drop Poincaré invariance. I mean in curved spacetime. There you may have uncountably many inequivalent representations of the same algebra of observables just by varying some continuous parameter (curvatures). If you put all these reps in orthogonal sectors in a common Hilbert space, it must be non.separable. Each sector is however separable.
However I do not see any cogent reason to use that overall space. The algebraic approach is easier to handle. Each single Hilbert-space sector is constructed by means of the GNS construction.
In flat spacetime it is not necessary to introduce uncountably many orthogonal sectors making the overall Hilbert space non-separable. In condensed matter I suspect that orthogonality and thus non.separability arises only taking the thermodynamical limit. So these sectors cannot co-exist simultaneously in our universe.