  An extended field theory is presented that captures the full SL(2) x O(6,6+n)
duality group of four-dimensional half-maximal supergravities. The theory has
section constraints whose two inequivalent solutions correspond to minimal D=10
supergravity and chiral half-maximal D=6 supergravity, respectively coupled to
vector and tensor multiplets. The relation with O(6,6+n) (heterotic) double
field theory is thoroughly discussed. Non-Abelian interactions as well as
background fluxes are captured by a deformation of the generalised
diffeomorphisms. Finally, making use of the SL(2) duality structure, it is
shown how to generate gaugings with non-trivial de Roo-Wagemans angles via
generalised Scherk-Schwarz ansaetze. Such gaugings allow for moduli
stabilisation including the SL(2) dilaton.
