Summary: We study the derived category of a complete intersection \(X\) of bilinear divisors in the orbifold \(\operatorname{Sym}^2\mathbb{P}(V)\). Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between \(\operatorname{Sym}^2\mathbb{P}(V)\) and a category of modules over a sheaf of Clifford algebras on \(\mathbb{P}(\operatorname{Sym}^2V^{\vee })\). The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating \(D^b(X)\) into a derived category of factorisations on a Landau-Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi-Yau 3-folds have equivalent derived categories.