It is certainly possible to express ordinary quantum mechanics in the framework of a real vector space: by adopting a suitable restriction on all operators--Stueckelberg’s rule--one can make the real-vector-space theory exactly equivalent to the standard complex theory. But can we achieve a similar effect without invoking such a restriction? In this talk I explore a model within real-vector-space quantum theory in which the role of the complex phase is played by a separate physical system called the ubit (for “universal rebit”). The ubit is a single binary real-vector-space quantum object that is allowed to interact with everything in the world. It also rotates in its two-dimensional state space. In the limit of infinitely fast rotation, one recovers standard quantum theory. When the rotation rate is large but not infinite, one finds small deviations from the standard theory. Here I describe a few such deviations that we have seen numerically and explained analytically.