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king vitamin

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I am curious about corresponding results for non-relativistic theories. Let's say I have a Euclidean field theory where there isn't full rotational symmetry; if a definite example is needed, let's say I want it to Wick-rotate to a Galilean-invariant theory. Is there a corresponding theorem for what properties are needed in the Euclidean theory for the Wick rotation (which is used all the time) to guarantees a valid Hermitian quantum field theory?

I don't have an extremely strong background in the mathematics behind AQFT. I did find this statement of the OS theorem by Urs Schreiber (who I know posts here), which gives rotational invariance as an assumption. So my question boils down to whether this assumption can be relaxed.