Abstract: We introduce a canonical way of performing the joint lift of a Brownian motion W and a low-regularity adapted stochastic process X, extending [Diehl, Oberhauser, and Riedel (2015). A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations]. Applying this construction to the case where X is a one-dimensional fractional Brownian motion (possibly correlated with W) completes the partial rough path of [Fukasawa and Takano (2024). A partial rough path space for rough volatility]. We use this to model rough volatility with the versatile toolkit of rough differential equations (RDEs), namely by taking the price and volatility processes to be the solution to a single RDE. The lead-lag scheme of [Flint, Hambly, and Lyons (2016). Discretely sampled signals and the rough Hoff process] is extended to our fractional setting as an approximation theory for the rough path in the correlated case. Continuity of the solution map transforms this into a numerical scheme for RDEs. We numerically test this framework and use it to calibrate a simple new rough volatility model to market data. This is joint work with Ofelia Bonesini (LSE), Emilio Ferrucci (Oxford) and Antoine Jacquier (Imperial College London).