  We present a prescription in F-theory for realizing matter in "exotic"
representations of product gauge groups. For 6D vacua, bifundamental
hypermultiplets are engineered by starting at a singular point in moduli space
which includes 6D superconformal field theories coupled to gravity. A
deformation in Higgs branch moduli space takes us to a weakly coupled gauge
theory description. In the corresponding elliptically fibered Calabi--Yau
threefold, the minimal Weierstrass model parameters $(f,g,\Delta)$ vanish at
collisions of the discriminant at least to order $(4,6,12)$, but with
sufficiently high order of tangency to ensure the existence of T-brane
deformations to a weakly coupled gauge theory with exotic bifundamentals. We
present explicit examples including bifundamental hypermultiplets of
$\mathfrak{e}_7 \times \mathfrak{su}_2$ and $\mathfrak{e}_6 \times
\mathfrak{su}_3$, each of which have dual heterotic orbifold descriptions.
Geometrically, these matter fields are delocalized across multiple points of an
F-theory geometry. Symmetry breaking with such representations can be used to
produce high dimension representations of simple gauge groups such as the
four-index symmetric representation of $\mathfrak{su}_2$ and the three-index
symmetric representation of $\mathfrak{su}_3$, and after further higgsing can
yield discrete non-abelian symmetries.
