The most recent discussion of what axioms one might drop from the Wightman axioms to allow the construction of realistic models that I'm aware of is Streater, Rep. Prog. Phys. 1975 38 771-846, "Outline of axiomatic relativistic quantum field theory". I'm looking for any more recent review that seriously discusses the axioms.
A critique of the Haag-Kastler axioms would also be welcome, but I would prefer to stay close enough to Lagrangian QFT to allow relatively immediate characterization of the difference between models of the reduced axiomatic system and the standard models that are relatively empirically successful.
I'm specially interested in any reviews that include a discussion of what models are possible if one relinquishes the existence of a lowest energy vacuum state (we know that, at least, this weakening allows the construction of thermal sectors of the free field, and that such a sector contains a thermal state that is thermodynamically stable even though it is not minimum energy, and that a Poincaré invariant "extra quantum fluctuations" sector is also possible—I'd like to know what is the full gamut of such models?).