We introduce the "Asymmetric Radiative Transfer In Shells Technique" (Artist), a new method for photon propagation on large scales that explicitly conserves photons, propagates photons at the speed of light, approximately accounts for photon directionality, and closely reproduces results of more detailed radiative transfer (RT) codes. Crucially, it is computationally fast enough to evolve the large cosmological volumes required to predict the 21cm power spectrum on scales that will be probed by future experiments targeting the Epoch of Reionisation (EoR). Most semi-numerical models aimed at predicting the EoR 21cm signal make use of an excursion set formalism (ESF) approach, which achieves computational viability by compromising on photon conservation, constraining ionised regions to be spherical by construction, and not accounting for light-travel time. By implementing our RT method within the semi-numerical code SimFast21, we show that Artist predicts a significantly different evolution for the EoR ionisation field compared to the code's native ESF. In particular, Artist predicts a more gradual evolution of the volume-averaged ionisation fraction, and up to an order-of-magnitude difference in the ionisation power, depending on the physical parameters assumed. Its application to large-scale EoR simulations will therefore allow more physically-motivated constraints to be obtained for key EoR parameters, such as the escape fraction. and moment-based methods, with different strengths and weaknesses, balancing accuracy versus computational efficiency (Trac & Gnedin 2011). Often, while these methods can be made optimally accurate with sufficient computational investment, they remain computationally prohibitive in large-scale cosmological simulations that seek to reproduce the evolution of the universe on at least tens of Mpc scales, while simultaneously ensuring that the injection and propagation of photons on the smallest scales is both accurate and self-consistent.One particular such case, currently at the forefront of astrophysical research, is the modelling of the last global phase-change in the history of the universe -the epoch of reionisation (EoR). The sources of the photons respon-