  Given a gauged linear sigma model (GLSM) $\mathcal{T}_{X}$ realizing a
projective variety $X$ in one of its phases, i.e. its quantum K\"ahler moduli
has a maximally unipotent point, we propose an \emph{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 Pl\"ucker embedding of the Grassmannian $G(k,N)$.
