  The patterns of fringes produced by an interferometer have long been
important testbeds for our best contemporary theories of physics. Historically,
interference has been used to contrast quantum mechanics to classical physics,
but recently experiments have been performed that test quantum theory against
even more exotic alternatives. A physically motivated family of theories are
those where the state space of a two-level system is given by a sphere of
arbitrary dimension. This includes classical bits, and real, complex and
quaternionic quantum theory. In this paper, we consider relativity of
simultaneity (that observers may disagree about the order of events at
different locations) as applied to a two-armed interferometer, and show that
this forbids most interference phenomena more complicated than those of complex
quantum theory. If interference must depend on some relational property of the
setting (such as path difference), then relativity of simultaneity will limit
state spaces to standard complex quantum theory, or a subspace thereof. If this
relational assumption is relaxed, we find one additional theory compatible with
relativity of simultaneity: quaternionic quantum theory. Our results have
consequences for current laboratory interference experiments: they have to be
designed carefully to avoid rendering beyond-quantum effects invisible by
relativity of simultaneity.
