Reflection positivity in general

In the Euclidean QFT obtained by "Wick-rotating" a unitary QFT, the correlation functions satisfy a property called reflection positivity, see e.g. this Wikipedia article for the case of a scalar field.

What's the correct formulation if you have chiral fermions and/or terms like the QCD theta angle? Could someone give the references?

To begin the discussion, let's just consider spacetime of the form $$Y \times \mathbb{R}_t$$. We have a Hilbert space associated to $$Y \times 0$$ and some operators $$\phi(y,0)$$ which we're interested in computing correlation functions with. We define the usual time evolved operators by $$\phi(y,t) = e^{-itH} \phi(y,0) e^{itH}.$$ These have the nice property of being Hermitian if $$\phi(y,0)$$ is. We can also define the analytically continued operators by just replacing $$t$$ with $$z = t + i\tau$$. Now taking the adjoint we find (assuming $$\phi(y,0)$$ is Hermitian) $$\phi(y,z)^\dagger = \phi(y,z^*),$$ in particular $$\phi(y,i\tau)^\dagger = \phi(y,-i\tau).$$
Now let us consider the states $$\phi(y,z)|0\rangle$$ obtained from the vacuum $$|0\rangle$$. These are nonzero Hilbert space states, so they have positive norm, ie. $$\langle 0 | \phi(y,z)^\dagger \phi(y,z)|0\rangle = \langle 0 | \phi(y,z^*) \phi(y,z) | 0 \rangle > 0.$$ (Note that this is automatically properly imaginary-time-ordered as long as $$\tau>0$$, which we need anyway to have good states). In particular $$\langle 0 | \phi(y,-i\tau) \phi(y,i\tau) | 0 \rangle > 0.$$ We see the reflection principle at work here, $$i\tau \mapsto -i\tau$$, so if we imagined this was computed in a Euclidean path integral, it would be a reflection-symmetric configuration on $$Y \times \mathbb{R}_{\tau}$$.