I agree that there is only one physical reality. I don't like the dualist view. Dualism with mind-independent reality is awkward since it does not allow us to describe the interaction of mind and matter (or whatever the two entities are). As soon as we are able to decsribe this interaction we are forced towards monism.
But nevertheless there seem to be different layers of what we call reality; one layer is our mind.
Look at the rainbow; you have three different "views":
(a) the rainbow as it is perceived
(b) the physical constituents of the rainbow = the raindrops, lighrays etc.
(c) the mathematical description in terms of geometry etc.
(a) is unthinkable without YOU!
(b) is rather clear for the rainbow but not known for the real quantum world; our all-days language is no longer suitable for describing the quantum reality
(c) does "exist" even if you and the raindrops does not.
Now let's assume for a moment that the mathematical Hilbert space is not only a description of the reality but IS the reality (bad idea: we immediately face the problem of the collapse of the wave funtion which is not described in terms of quantum mechanics; so this reality is incomplete). In that case we agree that there is one reality which IS the Hilbert space - done!
I think the major step forward is that we agree that there is a reality and that we do not insist on idealism or positivism. The problem then is that we do not know what this reality IS.
If we would instead think of the q.m. formalism as a pure description only and if we say that there IS nothing else but this description (no external reality), then we omit the measurement problem, but we are left with the situation that after all we cannot answer the question "why do all physicists use the same description?".
In the very end I believe that every physicist tends to realism. After all calculations are done and after the experimental data have been analyzed he/she wants to know what there really IS (including himself/herself)