  Tambara and Yamagami investigated a simple set of fusion rules with only one
non-invertible object, and proved under which circumstances those rules could
be given a coherent associator. We consider a generalization of such fusion
rules to the setting where simple objects are no longer required to be split
simple. Over the real numbers, this means that objects are either real,
complex, or quaternionic. In this context, we prove a similar categorification
result to the one of Tambara and Yamagami.
