CMIIW, but as I understand it, Wick rotation replaces the Minkowski basis (t,x,y,z) with the Euclidean basis (it,x,y,z). Suppose that $t_2=t_1 cosh \beta+x_1 sinh \beta$ and $x_2=t_1 sinh \beta+x_1 cosh \beta$. If a Euclidean basis is defined with $\tau_1=i t_1$, then a coordinate vector with a real value $\tau_1 \neq 0$ will have an imaginary component $t_1$ and a real component $x_1$. However, converted to $t_2$ and $x_2$, this separation of real and imaginary coordinates doesn't apply.
So even if two coordinate frames have the same origin and the same tangent spacetime, their Wick-rotations span different spaces. Is there a Cauchy-like criterion that would make this redundant? If so, how does it follow from known physics? If not, does using Wick rotation imply postulating complex space?