= Global obstruction to Wick rotation of conformal compactifications
The <Projective Minkowski compactification> has infinite cyclic fundamental group, while the <conformal compactification of Euclidean space> is $S^4$ and simply connected. Thus the affine substitution $t\mapsto it$ cannot extend to an invertible smooth identification of these real compactifications. Both are real slices of the same complex <Klein quadric>, which permits local analytic continuation without equating their global real topologies.
In affine cone coordinates $[U,V,t,\boldsymbol x]$ with $2UV+t^2-|\boldsymbol x|^2=0$ and $U=1$, the substitution is represented on the complex quadric by the rational map $[U,V,t,\boldsymbol x]\mapsto[U^2,UV+t^2,iUt,U\boldsymbol x]$. It preserves the quadric equation but is indeterminate at some infinity points, including $[0,1,0,\boldsymbol0]$. This is a direct algebraic manifestation of the obstruction to a global regular Wick map.
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