  We discuss the possible realisation in string/M theory of the recently
discovered family of four-dimensional maximal $SO(8)$ gauged supergravities,
and of an analogous family of seven-dimensional half-maximal $SO(4)$ gauged
supergravities. We first prove a no-go theorem that neither class of gaugings
can be realised via a compactification that is locally described by ten- or
eleven-dimensional supergravity. In the language of Double Field Theory and its
M theory analogue, this implies that the section condition must be violated.
Introducing the minimal number of additional coordinates possible, we then show
that the standard $S^3$ and $S^7$ compactifications of ten- and
eleven-dimensional supergravity admit a new class of section-violating
generalised frames with a generalised Lie derivative algebra that reproduces
the embedding tensor of the $SO(4)$ and $SO(8)$ gaugings respectively. The
physical meaning, if any, of these constructions is unclear. They highlight a
number of the issues that arise when attempting to apply the formalism of
Double Field Theory to non-toroidal backgrounds. Using a naive brane charge
quantisation to determine the periodicities of the additional coordinates
restricts the $SO(4)$ gaugings to an infinite discrete set and excludes all the
$SO(8)$ gaugings other than the standard one.
