This question is a bit open-ended for my taste on MathOverflow, but it's not so bad as to not deserve and answer. Yes, of course, there is much research on mathematical rigor in quantum field theory. Of course, I don't know what "reasonable", "essentially different", and "realistic" mean to you, but I would say that there are "reasonable" approaches and that some of them do address "realistic" field theories. As an aside, "rigor" for its own sake is far from the primary goal of mathematical physics, as has been discussed many times here and on our sister sites. See for example http://theoreticalphysics.stackexchange.com/questions/107/the-role-of-rigor.
But, in any case, you mention QED, and more generally "realistic" quantum field theory almost certainly means Yang–Mills Theory + matter, as this is what appears in the Standard Model. Here there are deep open questions, like those related to the mass gap. But some parts are by now understood. One part in particular is the perturbative path-integral approach to Lagrangian field theory, where the deepest part of the story is that of renormalization theory. Kevin Costello's book does a good job, I think, of explaining to mathematicians what renormalization theory is, setting it within a language of homological algebra.
But you bring up Wightman's axioms and AQFT, suggesting that you are less interested in making rigorous physics-as-it-is-practiced, and more interested in axiomatizing its general structure. There is, of course, no consensus yet as to the correct axiomatization — almost all proposals have no nontrivial examples — but many structures seem to be common. There is a more flexible version of AQFT, called factorization algebra and detailed in Costello's book with Owen Gwilliam that I think goes a long way towards providing a basic framework. Certainly all physical theories will have more common structure than just being a factorization algebra; but it is very common that we write down an axiom system and then the examples that appear "in nature" are quite special.
I have yet to be convinced that the "Schrodinger" picture is a fundamentally correct one. This is the picture that underscores, for example, geometric quantization, and also the Atiyah–Kontsevich–Segal–etc picture of QFT (originally TQFT, but by now more general). Certainly this version of QFT is an interesting mathematical structure to study, but it arises from the "Heisenberg" picture underpinning factorization algebras only because sometimes certain algebraic objects have unique, or at least canonical, irreducible projective representations.
Finally, I want to make one more comment concerning the success of perturbative and some nonperturbative QFT. Namely, almost every QFT that has been written down has been described in "path integral" formalism in some sense or another. Certainly this is true of the perturbative field theories in Kevin's book. But probably "having a path-integral description" is not fundamental to the notion of QFT. In a related way, "having a classical limit" is not fundamental. I highly recommend Dijkgraaf's Les Houches notes; this point is made in Section 2.2.