  We explore a model of the world based on real-vector-space quantum theory. In
our model the familiar complex phase appearing in quantum states is replaced by
a single binary object that we call the ubit, which is not localized and which
can interact with any object in the world. Ordinary complex-vector-space
quantum theory can be recovered from this model if we simply impose a certain
restriction on the sets of allowed measurements and transformations
(Stueckelberg's rule), but in this paper we try to obtain the standard theory,
or a close approximation to it, without invoking such a restriction. We look
particularly at the effective theory that applies to a subsystem when the ubit
is interacting with a much larger environment. In a certain limit it turns out
that the ubit-environment interaction has the effect of enforcing
Stueckelberg's rule automatically, and we obtain a one-parameter family of
effective theories--modifications of standard quantum theory--that all satisfy
this rule. The one parameter is the ratio s/omega, where s quantifies the
strength of the ubit's interaction with the rest of the world and omega is the
ubit's rotation rate. We find that when this parameter is small but not zero,
the effective theory is similar to standard quantum theory but is characterized
by spontaneous decoherence of isolated systems.
