  The splitting processes of bremsstrahlung and pair production in a medium are
coherent over large distances in the very high energy limit, which leads to a
suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. In this paper,
we continue analysis of the case when the coherence lengths of two consecutive
splitting processes overlap (which is important for understanding corrections
to standard treatments of the LPM effect in QCD), avoiding soft-gluon
approximations. In particular, this paper analyzes the subtle problem of how to
precisely separate overlapping double splitting (e.g.\ overlapping double
bremsstrahlung) from the case of consecutive, independent bremsstrahlung (which
is the case that would be implemented in a Monte Carlo simulation based solely
on single splitting rates). As an example of the method, we consider the rate
of real double gluon bremsstrahlung from an initial gluon with various
simplifying assumptions (thick media; $\hat q$ approximation; large $N_c$; and
neglect for the moment of processes involving 4-gluon vertices) and explicitly
compute the correction $\Delta\,d\Gamma/dx\,dy$ due to overlapping formation
times.
