We discuss a fundamental property of open quantum systems: the quantum phases associated with their dynamical evolution are non-additive. We develop our argument by considering a multiplepath atom interferometer in the vicinity of a perfectly conducting plate. The coupling with the environment induces dynamical corrections to the atomic phases. In the specific example of a Casimir interaction, these corrections reflect the interplay between field retardation effects and the external atomic motion. Non-local open-system Casimir phase corrections are shown to be nonadditive, which follows directly from the unseparability of the influence functional describing the coupling of the atomic waves to their environment. This is an unprecedented feature in atom optics, which may be used in order to isolate non-local dynamical Casimir phases from the standard quasi-static Casimir contributions.Open quantum systems [1,2] have motivated a worldwide theoretical and experimental research effort. Basic quantum phenomenon such as decoherence [3] have been reported in a variety of mesoscopic systems. Atom interferometers [4] in the vicinity of a conducting surface constitute a particulary relevant and rich class of open quantum systems, in which both long-lived (atomic dipole) and short-lived (electric field) degrees of freedom are simultaneously at work.Here, we propose to use this example in order to demonstrate an unprecedented, key property of open quantum systems: the non-additivity of the quantum phases arising from their dynamical evolution. The coupling to the environment is described by an influence functional [5] depending simultaneously on a pair of quantum paths. This stands in sharp contrast to the quantum phases resulting from a unitary evolution, which depend only on single paths taken separately. While the phase differences associated to single-path contributions are additive by construction, the additivity has no reason to be valid for the influence functional phases associated to pairs of paths. In general, the doublepath influence functional phases cannot be separated into sums of single-path contributions [1]. The non-additivity of the environment-induced quantum phases is a direct consequence of this unseparability, which is intimately connected to the non-locality of these phases.Atom interferometers have been used to probe atomsurface interactions in the van der Waals (vdW) regime [6,7], turning atom optics into a promising field for the experimental investigation of dispersive forces [8][9][10][11][12][13]. The effect of surface interactions onto atomic waves propagating near a conducting plate is commonly described by means of the vdW (or Casimir-Polder at longer distances) potential taken at the instantaneous atomic position. In this description, the external atomic waves are treated as a closed quantum system driven by conservative forces.Nevertheless, we have shown recently [14] that such an approach is incomplete. This is so, because the external atomic degrees of freedom (d.o.f.s) are coupled to the ...