The design and verification of distributed software systems is often hindered by their ever-increasing complexity and their asynchronous operational semantics. This article considers choreography specifications for distributed systems to reduce that complexity. We use labelled state-transitions systems as ground model for both choreographies and the corresponding distributed systems. Based on Event-B method, we propose a stepwise correct-by-construction model to build asynchronous distributed systems which a priori realise their choreographies. We rely on a sufficient and necessary realisability condition and we apply several refinement steps w.r.t. that condition to generate the distributed peers. The first refinement returns peer behaviours obtained by synchronous projection. The previously computed system is then refined into its asynchronous version using unbounded FIFO buffers. We prove, thanks to invariant preservation, that a sequence of exchanged messages is preserved at each refinement step. We provide a formalised proof of a realisability algorithm for deterministic choreographies. Besides that, our contribution is twofold: the approach is a priori and the problackposed solution scales up to any number of peers communicating with each other.