The Rarita-Schwinger action in curved $n$-dimensional spacetime is
$$ \int \sqrt{g} \overline{\psi}_a \gamma^{abc} D_b \psi_c $$
Here $g = \det(g_{\mu \nu})$, and the indices $a, b \dots$ are 'internal' indices that transform under e.g. $\mathrm{SO} (3,1) $ in $3+1$ dimensions. $\gamma^{abc} = \gamma^{[a} \gamma^{b} \gamma^{c]}$ with the gamma matrices obeying $\gamma^a \gamma^b + \gamma^b \gamma^a = 2 \eta^{ab} $, and $\eta^{ab}=\mathrm{diag}(1,1 \ldots 1,-1,-1 \ldots -1)$ is the 'internal metric'. $\psi_{\mu} = \psi_{c} e^{c}_{\mu} $ is a spinor-valued one form. Spacetime indices $\mu, \nu$ can be 'converted' to internal indices using the frame field $e_a^{\mu}$, and vice versa. The covariant derivative is $D_{\mu} \psi_{\nu} =\partial_{\mu} \psi_{\nu} + \frac{1}{4} \omega_{\mu}^{ab} \gamma_{ab} \psi_{\nu} $. Here $\omega$ is taken to be the torsion free spin connection, and $\gamma^{ab} = \gamma^{[a} \gamma^{b]}$.
In flat space, the covariant derivative becomes a normal derivative, and the action then has a symmetry $\psi_c \rightarrow \psi_c + \partial_c \phi $, with $\phi$ an arbitrary function. This freedom can be used to eliminate some of the degrees of freedom from the field $\psi_c$ which correspond to lower spin. However, in curved space there is no corresponding symmetry under $\psi_c \rightarrow \psi_c + D_c \phi$. For this reason, it is said that the Rarita-Schwinger action in curved spacetime is inconsistent. My question is, what goes wrong when you don't have this extra symmetry? And do the problems manifest at the classical level or only at the quantum level?
