When linearizing binary quadratic programs, the most usual way is to replace bilinear products with additional variables constrained to take on consistent values in any feasible solution. In this setting, column generation is a principally desirable solution technique, for instance because the number of such additional linearization variables may be large while many of them may turn out to be zero in an optimal solution. Even more, the consistency constraints of the most customary linearization technique add significantly to the degeneracy of the basic solutions traversed when solving the continuous relaxation with the simplex algorithm. At the same time, solving this relaxation by column generation poses difficulties. The foremost one is that essentially all linearization variables with a negative objective coefficient need to be incorporated in order to deduce a valid lower bound for minimization from such a procedure in general, with only rare exceptions where an actual decreasing effect on the objective might be ruled out by other means than reoptimization. This translates accordingly to positive coefficients for upper bounds when maximizing the objective. To resolve this, we present a column generation approach built upon a strategic combination of classical techniques such as boolean complementation and continuous Rhys forms in order to maintain dual feasibility and thus to provide a valid dual bound at any time.