Consistent Quantum States over Spacetime without a Common Quantum Process

Determining whether multiple record-conditioned quantum descriptions admit a single underlying process is a central consistency problem for quantum states over spacetime (QSOSTs) and relational quantum descriptions. Yet causally agnostic interferometry accesses only a lower-dimensional QSOST projection, which inevitably leaves some process degrees of freedom unresolved. We combine branch-resolved QSOST interferometry with positive-process lifting and semidefinite convex duality to characterize this gluing problem. We prove that every positive-weight QSOST branch has a unique least positive lift and reduce both settingwise and common-process realizability to deterministic-process domination. Using this criterion, we construct a minimal bipartite-qubit example with two record settings and two outcomes in the definite causal order $A\prec B$: each setting is separately realizable, and both share the same complete unconditional QSOST, yet no single deterministic process realizes them jointly. The separation has the exact visibility threshold $\eta =1/\sqrt{2}$ , admits a sparse witness involving eight interferometric quadratures, and persists for full-rank processes. These results establish an operational boundary between QSOST consistency and global process consistency, and provide a practical framework for testing whether multiple quantum perspectives can share one standard-quantum process.