We present a novel method for the generation of realistic white matter numerical phantoms with controllable morphology, and more realistic orientation-dispersion and packing-density. A shift of paradigm is proposed: rather than ‘packing fibres’, our algorithm ‘grows fibres’ contextually and efficiently, avoiding intersection between fibres. The potential of the method is demonstrated by reaching the highest orientation-dispersion and packing-density ever. The flexibility of the method is demonstrated with simulations of diffusion-weighted MR signal in three example substrates with differing orientation-dispersions, packing-densities and permeabilities. A proof-of-concept application is presented to investigate the impacts of undulation and beading on the intra-axonal axial diffusivity.