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 $[U,V,t,\mathbf{x}]$ with $2UV+t^2-|\mathbf{x}{|}^2=0$ and $U=1$, the substitution is represented on the complex quadric by the rational map $[U,V,t,\mathbf{x}]\mapsto [U^2,UV+t^2,iUt,U\mathbf{x}]$. It preserves the quadric equation but is indeterminate at some infinity points, including $[0,1,0,0]$. This is a direct algebraic manifestation of the obstruction to a global regular Wick map.