  We classify the RBA-bases of $6$-dimensional noncommutative semisimple
algebras for which the algebra has a positive degree map. We show that these
RBAs are parametrized by seven real numbers, the first four of which are
positive and the remaining three arbitrary. Our classification gives formulas
for their standard bases and structure constants. Using these we give a list of
all noncommutative integral table algebras of rank 6 with order up to 150. Four
in the list are primitive, but we show these cannot be realized as adjacency
algebras of association schemes. In the last section of the paper we apply our
methods to give a precise description of the noncommutative integral table
algebras of rank 6 for which the multiplicity of both linear characters is 1.
