QED as a Wightman theory of observable fields?  With a collision theory? [Note: I'm using QED as a simple example, despite having heard that it
is unlikely to exist.  I'm happy to confine the question to
perturbation theory.]
The quantized Aᵘ and ψ fields are non-unique and unobservable.  So,
two questions:
A.  Can we instead define QED as the Wightman theory of Fᵘᵛ, Jᵘ, Tᵘᵛ,
and perhaps a handful of other observable, physically-meaningful
fields?  The problem here is to insure that the polynomial algebra of
our observable fields is, when applied to the vacuum, dense in the
zero-charge superselection sector.
B.  Is there a way to calculate cross sections that uses only these
fields?  This might involve something along the lines of Araki-Haag
collision theory, but using the observable fields instead of
largely-arbitrary "detector" elements.  (And since we're in the zero-charge
sector, some particles may have to be moved "behind the moon", as
Haag has phrased it.)
(Of course, the observable fields are constructed using Aᵘ and ψ.
But we aren't obliged to keep Aᵘ and ψ around after the observable
fields have been constructed.)
I suspect that the answers are: No one knows and no one cares.  That's
fine, of course.  But if someone does know, I'd like to know too.
[I heartily apologize for repeatedly editting this question in order
to narrow its focus.]
 A: Suppose that QED exists in the strongest feasible sense. This means that appropriately smeared fields in $A_\nu$ and $\psi$ with compact support are self-adjoint operators on some Hilbert space with a common dense nuclear domain, such that the operators (anti)commute for spacelike separated smeared fields, and formal expansion of the time-ordered correlation functions reproduces the standard perturbation expansion.
In this case, the gauge invariant even polynomial expressions of degree at most two are Wightman fields defining the vacuum sector of QED, and they generate a $C^*$-algebra satisfying the Haag-Kastler axioms. This is the observable subalgebra of the field algebra. 
As photons are massless, the standard Haag-Ruelle collision theory is not applicable, and as charged fields are missing, the scattering theory is not asymptotically complete. To get an asymptotic completion one would have to proceed in a DHR-like fashion and reconstruct intertwiners between the (uncountably many) superselection sectors of the theory. But DHR assumes a mass gap, hence the theory is not applicable. Nevertheless, if QED exists, the intertwiners exist, too, and are in fact heuristically known. However, the asymptotic charges states (electrons) are only infraparticles, as they (unlike bare electrons) carry their own elecromagnetic field. An asymptotic scattering theory of relativistic infraparticles (which should involve coherent state superselection sectors) has not been worked out so far. 
But work by Derezinski treats the nonrelativistic case rigorously, and work by Kulish and Faddeev indicates nonrigorously that nothing should go wrong in the relativistic case.
Thus a lot is known about how things should look like, but in the relativistic case there are neither constructions nor proofs. The best that has been done rigorously (by Salmhofer, I believe) is to construct QED as a field theory whose fields are formal power series in the coupling constant, but this is far from what is needed.
A: I cannot claim to be an expert on AQFT, but the parts that I'm familiar with rely on local fields quite a bit.
First, a clarification. In your question, I think you may be conflating two ideas: local fields ($\phi(x)$, $F^{\mu\nu}(x)$, $\bar{\psi}\psi(x)$, etc) and unobservable local fields ($A_\mu(x)$, $g_{\mu\nu}(x)$, $\psi(x)$, etc).
Local fields are certainly recognizable in AQFT, even if they are not used everywhere. In the Haag-Kastler or Brunetti-Fredenhagen-Verch (aka Locally Covariant Quantum Field Theory or LQFT), you can think of algebras assigned to spacetime regions by a functor, $U\mapsto \mathcal{A}(U)$. These could be causal diamonds in Minkowski space (Haag-Kastler) or globally hyperbolic spacetimes (LCQFT). You can also have a functor assigning smooth compactly supported test functions to spacetime regions, $U\mapsto \mathcal{D}(U)$. A local field is then a natural transformation $\Phi\colon \mathcal{D} \to \mathcal{A}$ between these two functors. Unwrapping the definition of a natural transformation, you find for every spacetime region $U$ a map $\Phi_U\colon \mathcal{D}(U)\to \mathcal{A}(U)$, such that $\Phi_U(f)$ behaves morally as a smeared field, $\int \mathrm{d}x\, f(x) \Phi(x)$ in physics notation.
This notion of smeared field is certainly in use in the algebraic constructions of free fields as well as in the perturbative renormalization of interacting LCQFTs (as developed in the last decade and a half by Hollands, Wald, Brunetti, Fredenhagen, Verch, etc), where locality is certainly taken very seriously.
Now, my understanding of unobservable local fields is unfortunately much murkier. But I believe that they are indeed absent from the algebras of observables that one would ideally work with. For instance, following the Haag-Kastler axioms, localized algebras of observables must commute when spacelike separated. That is impossible if you consider smeared fermionic fields as elements of your algebra. However, I think at least the fermionic fields can be recovered via the DHR analysis of superselection sectors. The issue with unobservable fields with local gauge symmetries is much less clear (at least to me) and may not be completely settled yet (though see some speculative comments on my part here).
A: The answer to "A" came from John Baez: Include all of the local
gauge-invariant fields in the theory.  And if that doesn't span the
zero-charge superselection sector, then I'm on the wrong track.
I will confess, though, that I personally wanted to construct the
zero-charge superselection sector by applying (the polynomial algebra
of) a finite handful of local observables fields to the vacuum.  I
want a finite set of defining fields.  (But I won't go into my
motivations here.)
Regarding "B", well, there does exist my own ancient work along these
lines:
http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-2843.pdf

The above is the (free) preprint for the Phys Rev D article:
http://prd.aps.org/abstract/PRD/v27/i6/p1340_1

But my work was nonrigorous.  And I didn't handle spin.  (I've fixed
that since.)  And it was computationally impractical.  But most
important, the incoming state of two colliding particles is obtained
by filtering an initial state that is required to include a piece that
describes the two colliding particles.  Finding such a state was left
at the "hunt and peck" stage -- a real weak point.  And this point
becomes weaker if, in "A", there is an infinitude of defining fields
for the theory.
So:
The above is the best answer I've got, but I was hoping for better.
