The anomaly problem sais that loop quantization destroys the classical symmetries of the Holst action and that LQG is inconsistent and not viable as a quantum theory of gravity.
There are several related issues.
The constraint algebra consists of three constraints Gauß G, diffeomorphism constraint D, Hamiltonian H (relict of timelike diffeomorphism due to spacelike foliation). The constraint algebra is non-Lie, the commutator of H[f] and H[g] with two testfunctions f and g involves structure functions instead of constants; a canonical treatment or solution is not known!
The constraints are not solved equivalently but step-wise. H cannot be solved at all (up to know); formal solutions are known but unphysical. When discretizing and using the Weyl algebra quantization the infinitesimal generators D is no longer defined; only finite diffeomorphisms can be defined. That is a first hint that the quantization affects the constraint algebra in an uncontrolled way.
A quantum version of H is still unknown (not unique) so its constraint algebra is not known (ambiguous), either! Thiemann's trick plus quantization / discretization results in a H that seems to be unphysical (does not create volume). Some steps in the quantization seem to be ad hoc. That is a second hint that the quantization affects the constraint algebra in an uncontrolled way.
The derivation of the LQG constraint algebra involves one time-gauge fixing before quantization. That is well-known from QED and QCD (temporal or Weyl gauge). All other gauge fixings are done after quantization. Attempts to check consistency w/o time-gauge using full constraint quantization a la Dirac generates 2nd class constraints for a consistent treatment / solution in LQG is still unknown (2nd class constraints modify the commutator relations). There are hints that unphysical gauge d.o.f. become dynamical gauge artifacts when quantizing incorrectly! (Alexandrov)
The way the constraints are implemented are far from natural. Time-gauge is applied by setting the d.o.f. = the field to zero. Other constraints are solved by applying them to states. But as infinitesimal diffeomorphisms fail to be defined after quantization this seems to be problematic (I do not know what is the current status but the way Thiemann implemented the constraints results in a rather strange, ultra-local topology). Sometimes constraints are implemented a la Gupta-Bleuler which is known to be problematic in non-abelian gauge theories. There is nothing explicitly wrong, but many constructions are not unique, ad hoc or seem to be strange.
There are hints that the operator algebra does not close after quantization. It only closed modulo constraints. The way these additional terms vanish depend in some sense on the above mentioned tricks. So this is a hint that quantization destroys the off-shell closure of the algebra and that the quantized theory contains gauge artifacts / unphysical gauge d.o.f.!
The algebra depends implicitly on the Immirzi parameter β which is still not fully understood and which introduces a quantization ambiguity. A natural choice β=i is not viable b/c it introduces reality conditions which are not fully understood in the canonical framework. For real β the Hamiltonian is awefully complicated. When looking at it from the spin foam perspective the simplicity constrains are somehow related to the reality conditions. But the simplicity constraints become second-class, affect the path integral measure which has not been worked out completely. All this indicates that we should not expect too many insights from the spin-foam approach (as far as I can see the path integral is never able to solve fundamental problems).
There have been some papers (by Perez?) studying the relation of discretization and quantization; and even discretization in the classical theory. The relation of spin foams and the canonical approach indicates that both approaches are somehow "singular limits" where curvature etc. is located edges / vertices. That could mean that "discretization + quantization" is one major problem. It is clear that discretization affects the constraint algebra in several ways, especially in the path integral approach, so this is another hint that inconsistencies are introduced in an uncontrolled way.
Unfortunately there is not one paper summarizing these issues. The spin foam community has - to a large extent - decided to work on physical applications instead of consistently defining their theory. The first paper I am aware of is Nicolai's "outside view" from 2005 and Thiemann's reply in his "inside view" from 2006. My feeling was always that Thiemann did not fully address all issues. Then there are Alexandrov's papers from 2010-2012 or so, the best summary I am aware of which discusses many of these issues in detail - unfortunately w/o finding a solution.
http://arxiv.org/abs/hep-th/0501114
http://arxiv.org/abs/hep-th/0608210
http://arxiv.org/abs/1009.4475
https://www.physicsforums.com/showthread.php?t=545596
https://www.physicsforums.com/showthread.php?t=544728
You should have a look at the two threads where we discussed Alexandrov's in-depth analysis of the anomaly problem in detail (but forget his own proposal which did not succeed)
One remark: it is often said that Alexandrov's papers are outdated b/c they do not take the EPRL/FK models into accout. As far as I can see this is wrong b/c these two approaches do neither address nor solve any of the above mentioned issues! The anomaly problem is still there.
Another remark: all consistency checks regarding graviton propagators, vertices, semi-classical limit etc. are irrelevant b/c they do not address the regime whe the anomaly would kill the theory.
I do not know whether the spinor/twistor approach (Wieland's papers) sheds new light on these issues. For me it seems that it's nothing else but the introduction of new variables which is in many ways equivalent to the standard approach (and therefore has the same problems). But perhaps I am overlooking something.
I do not know whether Thiemann's dust approach will help; as far as I can see there is no progress regarding these anomaly related issues from the Erlangen group.
