We introduce a generative model for protein backbone design utilizing geometric products and higher order message passing. In particular, we extend a state-of-the-art model for protein backbone generation, FrameFlow, and represent the frames of the protein backbone as elements of the projective geometric algebra. This enables to use geometrically more expressive bilinear geometric products as paradigm for higher order message passing. The proposed model achieves high designability and diversity, while also sampling protein backbones that closely follow the statistical distribution of secondary structure elements found in naturally occurring proteins, a property so far only insufficiently achieved by state-of-the-art generative models.