In view of Haag's Theorem, it seems the Hilbert spaces of a free theory and an interacting theory are not the same. Though it seems very believable, I could not find a result that states that this is the case for all theories.
Intuitively, it seems obvious that the Hilbert space of an interacting theory involving distinct fields is not the same as any of the Hilbert spaces of the individual free theories. However, what if we consider the scalar field with $\phi^4$ interactions? In this case I was unable to convince myself that the Hilbert space of the interacting theory must be different than the Hilbert space of the free $\phi$ field.
Is it always the case that Hilbert space of an interacting theory is different than any free theory?