Classical tensor field theories have a PT theorem, so what changes in a QFT to require charge conjugation to be a part of the theorem? Charge conjugation seems a bit unrelated to space-time, but is an integral part of the theorem.
I have a suspicion this has to do with the Grassmann algebra of fermions, if this is the case, then would a purely bosonic QFT have a PT theorem?
EDIT: Robin gives a counter-example of this idea below, so it must be another aspect of QFT.
I am not very familiar with details of the proof of the $CPT$ theorem, but could it be that $T$ is anti-unitary? For example consider a bosonic QFT with a Klein-Gordon field $\phi$ and a vector field $A^\mu$, and take the interaction Lagrangian $$\mathcal L_\text{int} = \frac{1}{M^2} \epsilon^{\mu\nu\sigma\rho} (\partial_\nu A_\mu) (\partial_\rho \phi) (\partial_\sigma \phi^\dagger).$$
Under $PT$, $\epsilon^{\mu\nu\sigma\rho}$ is unchanged, but $\phi \leftrightarrow \phi^\dagger$ because $PT$ is anti-linear. Thus we need the anti-linear $C$, which also switches $\phi \leftrightarrow \phi^\dagger$, to make $\mathcal L_\text{int}$ invariant.
