  We study the geometry of type II supergravity compactifications in terms of
an oriented vector bundle $E$, endowed with a bundle metric of split signature
and further datum. The geometric structure is associated with a so-called
generalised $G$-structure and characterised by an $E$-spinor $\rho$, which we
can regard as a differential form of mixed degree. This enables us to
reformulate the field equations of type II supergravity as an integrability
condition of type $d_H\rho=0$, where $d_H=d+H\wedge$ is the twisted
differential on forms. Finally, we investigate some geometric properties of
integrable structures and formulate various no-go theorems.
