There are various levels of generality you can aim for when proving the spin statistics theorem but it's probably best to start with free fields coming from the Dirac action
\begin{equation}
S = \int i\bar{\psi} \gamma^\mu \partial_\mu \psi - m \bar{\psi} \psi \, dx.
\end{equation}
The canonical quantization procedure then tells us to define a conjugate momentum
\begin{equation}
\pi_\alpha = \frac{\delta L}{\delta \partial_0 \psi^\alpha} = i \psi^\dagger_\alpha.
\end{equation}
(If the spinors are Majorana there is a $\frac{1}{2}$ from Dirac brackets but this doesn't change the conclusion.) Now why do we have to impose $\left \{ \psi^\alpha(\textbf{x}), i\psi^\dagger_\beta(\textbf{y}) \right \} = i\delta^\alpha_\beta \delta(\textbf{x} - \textbf{y})$? Well let's try imposing $\left [ \psi^\alpha(\textbf{x}), i\psi^\dagger_\beta(\textbf{y}) \right ] = i\delta^\alpha_\beta \delta(\textbf{x} - \textbf{y})$ instead and seeing what this says about the creation an annihilation operators. We know the fundamental field needs to have one $a$ and one $b^\dagger$ to make the energy bounded from below.
\begin{equation}
\psi(x) = \int a(\textbf{p})^r u_r(\textbf{p}) e^{-ipx} + b^\dagger(\textbf{p})^r v_r(\textbf{p}) e^{ipx} \, \frac{d\textbf{p}}{(2\pi)^3}
\end{equation}
Along with this expression, we also need to know how to contract the plane wave solutions to the Dirac equation.
\begin{equation}
\bar{u}_r(\textbf{p}) u_s(\textbf{p}) = 2m \delta_{rs}, \;\; \bar{v}_r(\textbf{p}) v_s(\textbf{p}) = -2m \delta_{rs}
\end{equation}
The algebra then leads to
\begin{align}
[a(\textbf{p})^r, a^\dagger(\textbf{q})^s] = \delta^{rs} \delta(\textbf{p} - \textbf{q}), \;\; [b(\textbf{p})^r, b^\dagger(\textbf{q})^s] = -\delta^{rs} \delta(\textbf{p} - \textbf{q})
\end{align}
from which we see that $b^\dagger(\textbf{p})_r \left | 0 \right >$ has negative norm. This problem goes away when we use ant-commutators instead.
What this demonstrates is that the Pauli principle is obeyed by half-integer spin fields because of unitarity. As with all no-go theorems, the exceptions are interesting. In particle physics, ghosts are able to get around this because they are an artifact of fixing the gauge, not something physical. In statistical physics, many theories get around this too because they are intrinsically Euclidean. Therefore we don't mind at all if the Wick rotated versions of them turn out not to be unitary.
