We continue the analysis of the geometry of generic Minkowski N = 1, D = 4
flux compactifications in M-theory using exceptional generalised geometry, including the
calculation of the infinitesimal moduli spaces. The backgrounds can be classified into two
classes: type-0 and type-3. For type-0, we review how the moduli arise from standard de
Rham cohomology classes. We also argue that, under reasonable assumptions, there are
no appropriate sources to support compact flux backgrounds for this class and so the only
solutions are in fact G2 geometries. For type-3 backgrounds, given a suitable ∂
0∂¯0
-lemma,
we show that the moduli can be calculated from a cohomology based on an involutive subbundle of the complexified tangent space. Using a simple spectral sequence we prove quite
generally that the presence of flux can only reduce the number of moduli compared with
the fluxless case. We then use the formalism to calculate the moduli of heterotic M-theory
and show they match those of the dual Hull-Strominger system as expected.