Variable splitting is an old but widely used technique whichaims at dividing an initial complicated optimization problem into simplersub-problems. In this work, we take inspiration from this variable splitting idea in order to build efficient Markov chain Monte Carlo(MCMC) algorithms. Starting from an initial complex target distribution,auxiliary variables are introduced such that the marginal distributionof interest matches the initial one asymptotically. In addition to havetheoretical guarantees, the benefits of such an asymptotically exact dataaugmentation (AXDA) are fourfold: (i) easier-to-sample full conditionaldistributions, (ii) possibility to embed while accelerating state-of-the-artMCMC approaches, (iii) possibility to distribute the inference and (iv)to respect data privacy issues. The proposed approach is illustrated onclassical image processing and statistical learning problems.