Distributed Phase Tags: A Locality-Preserving Realification Compiler for Misty-State Quantum Programs

Complex amplitudes can be simulated over real Hilbert spaces by adjoining a two-dimensional real/imaginary degree of freedom. In a multipartite program, however, one globally shared phase tag is not a locality-preserving program transformation: every party would have to act on the same auxiliary system. We recast the distributed real simulation of multipartite quantum mechanics as a syntax-directed compiler into a real misty-state intermediate language. For n parties we give explicit phase-reference codewords 0 P n and 1 P n on n local tags and prove that the same real operator J=XZ on any single tag realizes the logical action of multiplication by i. If a complex party-local gate is A=A R+iA I, its real translation is [ A j=A R I+A I J j,] which acts only on party j 's data and local phase tag. We prove an intertwining theorem for arbitrary finite party-local pure-state circuits, exact preservation of computational-basis measurement probabilities, preservation of the run-time interaction graph, and one-rebit-per-party overhead. For the canonical three-party GHZ program, the translation lies entirely inside the ordinary black-and-white misty gate fragment: controlled- J replaces the complex phase operation, while the invariant J jJ k0 P=-0 P transfers the required branch sign. This yields a short correctness proof for all four promised GHZ inputs and explains the earlier 64-path cancellation calculation at a single logical phase-code boundary. We distinguish this program realification problem from the weaker task of finding any real strategy with the same correlations, and state precisely which parts are known real-Hilbert-space simulation and which parts belong to the misty-language compilation layer developed here.