  We use exceptional field theory as a tool to work out the full non-linear
reduction ansaetze for the AdS$_5\times S^5$ compactification of IIB
supergravity and its non-compact counterparts in which the sphere $S^5$ is
replaced by the inhomogeneous hyperboloidal space $H^{p,q}$. The resulting
theories are the maximal 5D supergravities with gauge groups SO(p,q). They are
consistent truncations in the sense that every solution of 5D supergravity
lifts to a solution of IIB supergravity. In particular, every stationary point
and every holographic RG flow of the scalar potentials for the compact and
non-compact 5D gaugings directly lift to solutions of IIB supergravity.
