In quantum field theories it is believed that anomalies in gauge symmetries (in contrast to rigid symmetries) cannot be coped with and must be canceled at the level of the elementary fields.
May be the earliest work on the subject is: C. Bouchiat, J. Iliopoulos and P. Meyer, “An Anomaly free Version of Weinberg’s Model” Phys. Lett. B38, 519 (1972).
But certainly, one of the most famous ones is the Gross-Jakiw article: Effect of Anomalies on Quasi-Renormalizable Theories Phys. Rev. D 6, 477–493 (1972)
They agrgued that the 'tHooft-Veltman perturbative proof of the renormalizability of gauge theories requires the anomalous currents not to be coupled to gauge fields.
In the more modern BRST quantization language, gauge anomalies give rise to anomalous terms in the Slavnov-taylor identities which cannot be canceled by local counter-terms therefore ruin the combinatorial proof of perturbative renormalizability and of the decoupling of the gauge components and ghosts which results a non-unitary S-matrix.
This phenomenon is independent of the quantum field theory dimension. In 3+1 dimension, the requirement of cancelation of the chirally coupled standard model leads to the correct particle content.
In 1+1 dimensions, it leads to the dimension of string target space (Virasoro anomaly) and the optional anomaly free gauged subgroups in the WZNW models in two dimensions (Kac-Moody anomaly).
In certain dimensions there is the Green-Schwrz Cancellation mechanism, but it is equivalent to an addition of a local term to the Lagrangian. This mechanism is not appropriate to 3+1 dimensions, since this term is not renormalizable.
The chiral anomaly can be corrected by adding a Wess-Zumino term to the Lagrangian, but this term is not perturbatively renormalizable, thus does not solve the nonrenormalizability problem.
However, the cancellation of anomalies does not mean that we should not seek for representations of the "anomalous" current algebras. At the contrary, in 1+1 dimensions the Virasora and the Kac-Moody algebra do not have positive energy highest weight representations unless they are anomalous.
According to this prionciple, the spectra of each sector of the theory are determined by the anomaly of this sector, in spite of the fact that the total anomaly vanishes.
In the late eighties, R.G. Rajeev and especially Juoko Mickelsson started a project in which they seeked representations of anomalous non-centrally extended algebras.(i.e., those present in 3+1 dimensions). The presence of gauge field depended Abelian extensions (The Mickelsson-Faddeev extension), made this problem hard to solve.
One route they adopted was to consider an algebraic universal gauge theory (in the same sense as universal classifying spaces). But a no-go result by Doug Pickrell stated that the universal anomalous algebra has no nontrivial unitary representations. This result is valid for the universal model but it is discouraging for the actual (Mickelsson-Faddeev) extended algebra. There are later works on the subject by Juoko Mickelsson himself and also by Edwin Langmann; but the question remains open.
Update:
Regarding the question about the validity of an anomalously gauged theory as an effective gauge theory (a la Weinberg, Let us consider the Skyrme model for definiteness): Both are strictly perturbatively non-renormalizable, but there is a big difference in their divergence behavior:
An anomalously gauged theory requires, in each order in its loop expansion counter-terms, with arbitrarily high derivatives (or arbitrarily high external momenta). In
contrast, due to the chiral perturbation theory of an effective field theory each order of the loop expansion requires only counter-terms with one order higher (by 2) in external momenta.
Chiral perturbation theory is unitary order by order in the momentum expansion. The bound on the powers of the momenta at each order makes it viable as a low energy effective theory. The current green functions respect non-anomalous Ward identities, which control the divergences.
In an anomalously gauged theory a truncation of the higher momentum power counter-terms would result S-matrix non-unitarity.
This is the main reason, why it is widely accepted that an anomalously gauged theory requires additional fields to cancel the anomalies. (This principle came into experimental demonstration in the prediction of the t-quark).
In regards of Drake's remark; there is the case of the chiral Schwinger model in two dimensions (2D-QED with chiral coupling of the electron to the photon). This model is exactly solvable. In the exact solution, the gauge boson acquires a mass which is an indication consistent with Drake's remark that more "Degrees of freedom are needed".
There is the "problem" that Drake pointed out, of how the "number of degrees of freedom" jumps between the (supposedly) non-anomalous classical theory and the quantum theory. My, point of view is that the space of the Grassmann variables describing the fermions (semi-)classically is not an appropriate phase space, although it is a symplectic supermanifold (Poisson brackets can be defined between Grasmann variables).
This difficulty was already encountered by: Berezin and Marinov, when they noticed that they cannot define a nontrivial Grassmann algebra valued phase space distribution, so, they stated that Grassmann variables acquire a meaning only after quantization.
One way to define a fermionic phase space on which a density of states can be defined is to pick the "manifold of initial data" as in the bosonic case. In finite systems, these types of manifolds turn out to be coadjoint orbits, however I don't know of any work in which the anomaly is derived classically on "the manifold of initial data" of a fermionic field. I think that on these "honest" fermionic phase spaces, chiral anomalies are present classically; however, I don't know of any work in this direction. I consider this a very interesting problem.
A remark on the non-uniqueness of the mechanism of anomaly cancellation: We can cancel anomalies by adding a new family of fermions, various Wess-zumino terms (corresponding to different anomaly free subgroups), and may be masses to the gauge fields (as in the Schwinger model).
This non-uniqueness, reflects the fact that when anomaly is present, the quantization is not unique, (in other words the theory is not completely defined). This phenomenon is known in many cases in quantum mechanics (inequivalent quantizations of a particle on a circle), and quantum field theory (theta vacua).
Finally, my point of view is that the anomaly cancellation does not dismiss the need of finding "representations" to the anomalous current algebras in each sector. This principle works in 1+1 dimensions. It should work in any dimension because according to Wigner quantum theory deals with representations of algebras. This is why I think that Mickelsson's project is important.
May be for higher dimensions, more general representations than representations on Hilbert spaces are needed due to the non-central extensions present in these dimensions. For me, this problem is very interesting.
