  Using the colour dipole picture and the colour glass condensate effective
theory, we study the diffractive production of two or three jets via coherent
photon-nucleus interactions at high energy. We consider the hard regime where
the photon virtuality and/or the transverse momenta of the produced jets are
much larger than the saturation momentum $Q_s$ of the nuclear target. We show
that, despite this hardness, the leading-twist contributions are controlled by
relatively large parton configurations, with transverse sizes $R\sim 1/Q_s$,
which undergo strong scattering and probe gluon saturation. For exclusive
dijets, this implies that both final jets have semi-hard transverse momenta
($P_\perp\sim Q_s$) and that one of them is aligned with the photon. The
dominant contributions to the diffractive production of hard dijets
($P_\perp\gg Q_s$) rather come from three-jet final states, which are very
asymmetric and will be referred to as 2+1 jets: two of the jets are hard, while
the third one is semi-hard. We demonstrate that the leading-twist contributions
to both exclusive dijets and the diffractive production of 2+1 jets admit
transverse-momentum dependent (TMD) factorisation, in terms of quark and gluon
diffractive TMD distribution functions, for which we obtain explicit
expressions from first principles. We show that the contribution of 2+1 jets to
diffractive SIDIS (semi-inclusive deep inelastic scattering) takes the form of
one step in the DGLAP evolution of the quark diffractive PDF.
