  We perform a case study of the behavior of gluon radiation beyond the soft
approximation, using as an example the Drell-Yan production cross section at
NNLO. We draw a careful distinction between the eikonal expansion, which is in
powers of the soft gluon energies, and the expansion in powers of the threshold
variable $1 - z$, which involves important hard-collinear effects. Focusing on
the contribution to the NNLO Drell-Yan K-factor arising from real-virtual
interference, we use the method of regions to classify all relevant
contributions up to next-to-leading power in the threshold expansion. With this
method, we reproduce the exact two-loop result to the required accuracy,
including $z$-independent non-logarithmic contributions, and we precisely
identify the origin of the soft-collinear interference which breaks simple
soft-gluon factorization at next-to-eikonal level. Our results pave the way for
the development of a general factorisation formula for next-to-leading-power
threshold logarithms, and clarify the nature of loop corrections to a set of
recently proposed next-to-soft theorems.
