Wick-connected theories and Lorentz violation

We consider double Wick rotation in field theories, which analytically continues the time coordinate, and then reinterprets one of the spatial directions as the new Lorentzian time. We show that if Lorentz-invariance is absent, Wick-connected theories are no longer necessarily equivalent. Focusing on flat spacetime, we study the propagating modes, unitarity and renormalizability of such Wick-connected theories, and provide criteria for when such notions fail to translate.