Twistor equations are invariant under conformal rescaling of metric and all geometric notions used in twistors are essentially conformally invariant. Twistors are reduced spinors of the pseudo-orthogonal group O(2,4) which happens to be locally isomorphic with conformal group of Minkowski space. So the description of space-time that you get using twistor coordinates enjoys the fifteen parameter conformal symmetry group, and not the ten parameter Poincare group. This explains why you need a compactified Minkowski space.

Twistor equations are invariant under conformal rescaling of metric and all geometric notions used in twistors are essentially conformally invariant. Twistors are reduced spinors of the pseudo-orthogonal group O(2,4) which happens to be locally isomorphic with conformal group of Minkowski space. So the description of space-time that you get using twistor coordinates enjoys the fifteen parameter conformal symmetry group, and not just the ten parameter Poincare group. This explains why we need a compactified Minkowski space.