= Solution
Let a <gauge transformation> act infinitesimally as $\delta A_\mu^a=(D_\mu\omega)^a$, and define the <Faddeev-Popov operator> by variation of the chosen gauge functional:
$$
\mathcal M^{ab}[A](x,y)=\left.\frac{\delta\mathcal F^a[A^\omega](x)}{\delta\omega^b(y)}\right|_{\omega=0}.
$$
Work in a perturbative <regular gauge slice>, with boundary conditions that remove residual zero modes and with one local representative of each <gauge orbit>. The <Faddeev-Popov gauge-orbit identity> is $1=\Delta_{\mathrm{FP}}[A]\int\mathcal D\omega\,\delta[\mathcal F[A^\omega]]$. Inserting it and factoring out the formal gauge-group volume gives
$$
\boxed{Z=\mathcal N\int\mathcal DA\,\delta[\mathcal F[A]]\det\mathcal M[A]\,e^{iS[A]}.}
$$
The functional delta imposes the gauge condition at every point and for every color. The <determinant> compensates the gauge-orbit Jacobian. For ordinary unoriented real gauge-orbit integration the local Jacobian is $|\det\mathcal M|$; perturbatively its sign is fixed and absorbed into $\mathcal N$, leaving the displayed <determinant>. Multiple global intersections, or a <Gribov ambiguity>, require more care; the local perturbative formula is not a global uniqueness assertion.
The complex <Grassmann Gaussian integral> now generalizes to spacetime and color indices. Introduce independent anticommuting <Faddeev-Popov ghost fields> $c^a(x),\bar c^a(x)$ and write
$$
\det\mathcal M[A]=\mathcal N'\int\mathcal D\bar c\,\mathcal Dc\,\exp\left(i\int d^dx\,d^dy\,\bar c^a(x)\mathcal M^{ab}[A](x,y)c^b(y)\right).
$$
The factors $i$ in the <functional determinant> differ from a real exponential only by a regulated, field-independent constant. These <Grassmann fields> are Lorentz scalars in the adjoint color representation; the antighost is independent of the ghost rather than an ordinary complex-conjugate commuting variable. They are unphysical fields, not external observable particles. Representing the <functional determinant> by a local ghost <action> when $\mathcal F$ is local makes ordinary <Feynman rules> available. Anticommutation gives the minus sign of a closed ghost loop, enabling cancellation of unphysical gauge contributions and maintaining the gauge identities in perturbation theory.
For example, the linear covariant choice $\mathcal F^a=\partial^\mu A_\mu^a$ gives $\mathcal M^{ab}=\partial^\mu D_\mu^{ab}$, so non-abelian ghost-gauge interactions remain. In an <axial gauge> $\mathcal F^a=n^\mu A_\mu^a$ with constant $n$, however,
$$
\mathcal M^{ab}=\delta^{ab}n\cdot\partial+gf^{acb}n\cdot A^c\quad\longrightarrow\quad\delta^{ab}n\cdot\partial
$$
on the exact gauge slice. Its <determinant> is field-independent, so \b[axial-gauge ghosts decouple] and contribute only a normalization factor. This statement uses the strict delta-functional constraint; a finite-width Gaussian gauge weight is not the same exact slice.
The disadvantage is that choosing $n$ obscures manifest Lorentz covariance and the <axial-gauge propagator> contains spurious $1/(n\cdot k)$ poles. An <axial-gauge pole prescription> and control of residual transformations satisfying $n\cdot\partial\omega=0$ are necessary; formal decoupling alone does not fix those problems.
\b[Abelian ghosts do not always decouple.] In a field-independent linear <Lorenz gauge>, $\delta A_\mu=\partial_\mu\omega$ gives $\mathcal M=\Box$ and they are free. But the <quadratic Abelian gauge fixing> $\mathcal F[A]=\partial\cdot A+\kappa A_\mu A^\mu$ gives
$$
\delta\mathcal F=(\Box+2\kappa A^\mu\partial_\mu)\omega.
$$
Its <Faddeev-Popov operator> depends on $A$, and the ghost <action> contains $2\kappa\bar c A^\mu\partial_\mu c$. This is an explicit abelian ghost-gauge interaction. Abelianity removes the commutator term in the <gauge transformation>, not the field dependence of an arbitrary nonlinear gauge condition.
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