  Non-global QCD observables are characterised by a sensitivity to the full
angular distribution of soft radiation emitted coherently in hard scattering
processes. This complexity poses a challenge to their all-order resummation,
that was formulated at the leading-logarithmic order about two decades ago. In
this article we present a solution to the long-standing problem of their
resummation beyond this order, and carry out the first complete next-to-leading
logarithmic calculation for non-global observables. This is achieved by solving
numerically the recently derived set of non-linear differential equations which
describe the evolution of soft radiation in the planar, large-$N_c$ limit. As a
case study we address the calculation of the transverse energy distribution in
the interjet rapidity region in $e^+e^-\to$ dijet production. The calculation
is performed by means of an algorithm that we formulate in the language of
jet-calculus generating functionals, which also makes the resummation technique
applicable to more general non-global problems, such as those that arise in
hadronic collisions. We find that NLL corrections are substantial and their
inclusion leads to a significant reduction of the perturbative scale
uncertainties for these observables. The computer code used in the calculations
is made publicly available.
