In this article we combine a recently proposed method for factorisation-aware matrix element surrogates with an unbiased unweighting algorithm. We show that employing a sophisticated neural network emulation of QCD multijet matrix elements based on dipole factorisation can lead to a drastic acceleration of unweighted event generation. We train neural networks for a selection of partonic channels contributing at the tree-level to Z + 4, 5 jets and t t̄ + 3, 4 jets production at the LHC which necessitates a generalisation of the dipole emulation model to include initial state partons as well as massive final state quarks. We also present first steps towards the emulation of colour-sampled amplitudes. We incorporate these emulations as fast and accurate surrogates in a two-stage rejection sampling algorithm within the SHERPA Monte Carlo that yields unbiased unweighted events suitable for phenomenological analyses and post-processing in experimental workflows, e.g. as input to a time-consuming detector simulation. For the computational cost of unweighted events we achieve a reduction by factors between 16 and 350 for the considered channels.