Given a gauged linear sigma model (GLSM) \({\mathcal {T}}_{X}\) realizing a projective variety X in one of its phases, i.e. its quantum Kähler moduli has a geometric point, we propose an extended GLSM \({\mathcal {T}}_{{\mathcal {X}}}\) realizing the homological projective dual category \({\mathcal {C}}\) to \(D^{b}Coh(X)\) as the category of B-branes of the Higgs branch of one of its phases. In most of the cases, the models \({\mathcal {T}}_{X}\) and \({\mathcal {T}}_{{\mathcal {X}}}\) are anomalous and the analysis of their Coulomb and mixed Coulomb-Higgs branches gives information on the semiorthogonal/Lefschetz decompositions of \({\mathcal {C}}\) and \(D^{b}Coh(X)\). We also study the models \({\mathcal {T}}_{X_{L}}\) and \({\mathcal {T}}_{{\mathcal {X}}_{L}}\) that correspond to homological projective duality of linear sections \(X_{L}\) of X. This explains why, in many cases, two phases of a GLSM are related by homological projective duality. We study mostly abelian examples: linear and Veronese embeddings of \({\mathbb {P}}^{n}\) and Fano complete intersections in \({\mathbb {P}}^{n}\). In such cases, we are able to reproduce known results as well as produce some new conjectures. In addition, we comment on the construction of the HPD to a nonabelian GLSM for the Plücker embedding of the Grassmannian G(k, N).