I will be happy if someone could clarify the mystery here.
Consider the following derivation of the anomalous Slavnov-Identity. It's based on lecture notes by Adel Bilal.
Suppose we have an action with a field (or fields) $\phi$, and a symmetry transformation which takes $$\phi \rightarrow \phi ' = \phi + \epsilon F(x, \phi(x)) ,$$ under which the action is invariant, $S[\phi '] = S[\phi]$.
However, let's suppose that the functional measure is anomalous: $${ \mathcal{D} \phi } \rightarrow { \mathcal{D} \phi } \space \space e^{i\epsilon \int \mathcal{A}(x)}.$$
Let's look at the following path integral (which is the generating functional of the connected diagrams):
$$ \int {\mathcal{D}\phi \space \text{exp}(iS[\phi] + i\int J(x)\phi(x))} = \int {\mathcal{D}\phi ' \space \text{exp}(iS[\phi '] + i\int J(x)\phi '(x))} = \int {\mathcal{D}\phi \space \text{exp} ({i\epsilon \int \mathcal{A}(x)} ) \space \text{exp}(iS[\phi] + i\int J(x)\phi(x) + i\epsilon \int J(x)F(x,\phi))} = \int {\mathcal{D}\phi \space \text{exp}(iS[\phi] + i\int J(x)\phi(x)) \left( 1 + i\epsilon \int A(x) + i\epsilon \int J(x)F(x,\phi)\right)}. $$
The first equality is just a change of integration variables, the second is the substitution of $\phi '$ in terms of $\phi$, and the third is just reorganization and first order expansion in $\epsilon$.
Now you get the Slavnov-Taylor identity (eq. 4.5 in the original paper):
$$ \int { (\mathcal{A}(x) + J(x) {\left< {F(x,\Phi)} \right>} _J) } = 0 \tag{4.5}$$
Can someone explain how this identity could be possible, since $\mathcal{A}$ doesn't depend on $J$, which is an auxiliary field, so substituting $J=0$ would give $ \int \mathcal{A} = 0$ which doesn't seem plausible.
The origin of the problem (if it is a problem as it seems) should be in the change of variables in the first equality above, but it seems valid. To simplify matters, I think the above equations could all be derived with $J=0$, to give:
$$ \int {\mathcal{D}\phi \space \text{exp}(iS[\phi])} = \int {\mathcal{D}\phi ' \space \text{exp}(iS[\phi '])} = \int {\mathcal{D}\phi \space \text{exp} ({i\epsilon \int \mathcal{A}(x)} ) \space \text{exp}(iS[\phi])} $$
Which doesn't seem plausible at all.
Maybe it's related to some regularization schemes that should usually be implemented in QFT, but it isn't immediate from these equations that you need any regularization here, so it's unclear to me what is wrong with these equations.