We study liftability as a global invariant of matter fields on spacetime. Fix a noninjective local diffeomorphism π: F → Qₜ and a smooth target-valued field Q. Although Q always admits local lifts, it need not admit a global lift. We prove a compactness theorem showing that, when the chronology order is future-directed and every chronological past has a finite nonempty set of lifts, a global lift must exist. Consequently, the phenomenon studied here cannot occur in Minkowski spacetime for finite-fiber presentations and requires horizon-type causal structure. We then construct a smooth field on a simply connected spacetime with de Sitter causal structure that admits a lift over the complete chronological past of every event but has no global lift. The phenomenon is realized both in an elementary two-branch model and with connected field space using the Pinchuk counterexample to the strong real Jacobian conjecture. We also prove a no-go result for bilaterally locally isomorphic worlds when lift failure has local witnesses, and show that continuation through a nonproper étale presentation can depend on the path itinerary rather than its homotopy class. The resulting obstruction is neither ordinary covering-space monodromy nor one of the standard degree-one or degree-two characteristic-class obstructions, all of which vanish on the simply connected spacetime used in the construction. Preprint, version 5. Independent expert verification remains outstanding. The stated Pinchuk sheet-pairing construction carries the explicitly identified Campbell Figure 3 verification gate.