  In previous works (arXiv:1610.07916, arXiv:1711.07921, arXiv:1807.00186) we
studied a class of toric Calabi-Yau threefolds which engineer six-dimensional
supersymmetric gauge theories with gauge group $U(N)$ and adjoint matter. The
K\"ahler moduli space of these manifolds can be extended through flop
transformations to include regions which are described by so-called shifted
toric web diagrams. In this paper we analyse gauge theories that are engineered
by these shifted toric web diagrams and argue that in specific limits, some of
the them engineer five-dimensional quiver gauge theories with gauge group
$G\subset U(N)$ and with fundamental and bi-fundamental matter. We discuss
several examples in detail and describe how the matter sector is obtained from
the six-dimensional parent theory.
