  We develop a new construction of complete non-compact 8-manifolds with
Riemannian holonomy equal to $\operatorname{Spin}(7)$. As a consequence of the
holonomy reduction, these manifolds are Ricci-flat. These metrics are built on
the total spaces of principal $T^2$-bundles over asymptotically conical Calabi
Yau manifolds. The resulting metrics have a new geometry at infinity that we
call asymptotically $T^2$-fibred conical ($AT^2C$) and which generalizes to
higher dimensions the ALG metrics of 4-dimensional hyperk\"ahler geometry,
analogously to how ALC metrics generalize ALF metrics. As an application of
this construction, we produce infinitely many diffeomorphism types of $AT^2C$
$\operatorname{Spin}(7)$-manifolds and the first known example of complete
toric $\operatorname{Spin}(7)$-manifold.
