quantum electrodynamics - Why is the spinor field anti-commutator not made gauge invariant?

When we introduce minimal coupling for the Dirac spinor field, we introduce terms into the Lagrangian, by the substitution $i\frac{\partial}{\partial x^\mu}\mapsto i\frac{\partial}{\partial x^\mu}+eA_\mu$, so that the Lagrangian is invariant under arbitrary changes of phase at every point of space-time, $$\psi(x)\mapsto e^{-ieG(x)}\psi(x),\qquad A_\mu\mapsto A_\mu+\partial_\mu G(x),$$ where $G(x)$ is a scalar function. We don't make this substitution, however, in the anti-commutator $$\left\{\psi_\xi(x),\overline{\psi_{\xi'}(x')}\right\}=\left(i\gamma^\mu\frac{\partial}{\partial x^\mu}+m\right)_{\xi\xi'}i\Delta(x-x'),$$ which, it seems, is therefore not invariant under the transformation $\psi(x)\mapsto e^{-ieG(x)}\psi(x)$, $\psi(x')\mapsto e^{-ieG(x')}\psi(x')$. Presumably the path-integral formalism doesn't care about this, but does this have consequences in subsequent derivations in the canonical approach?

I expect that in fussing about this I'm missing something very obvious. Am I? What is it?

EDIT: So it appears from the Answers, for which Thanks, that the anti-commutator given is valid for both space-like and time-like separation for the free Dirac field, and it's valid for space-like separation in QED, but it's nae valid for time-like separation in QED. Put somewhat loosely, this is OK in the canonical formalism because this anti-commutator is valid for the phase space. I'd rather like to know what a valid expression for the anti-commutator is at time-like separation in QED, but since that would, in a sense, be a solution of the theory I guess I'll have to whistle for it. I'll be grateful for Comments if this is an obviously obtuse reading of the Answers, though I ask that you consider that I would like a manifestly covariant version (so to speak, perhaps obscurely) of the canonical formalism before leaping in.

Answer

Because $\left(i\gamma^\mu\frac{\partial}{\partial x^\mu}+m\right)_{\xi\xi'}i\Delta(x-x')$ is a symbolic expression for a given analytical right-hand side. It is written so for convenience (not yet calculated) but it is a specific expression like $\delta(x-x')$ or $\delta(x-x')'$. It should not acquire any "gauge extension" by definition. This expression does not contain a "particle momentum".