We address the problem of reachability in distributed systems, modelled as networks of finite automata and propose and prove a new algorithm to solve it efficiently in many cases. This algorithm allows to decompose the reachability objective among the components, and proceeds by constructing partial products by lazily adding new components when required. It thus constructs more and more precise over-approximations of the complete product. This permits early termination in many cases, in particular when the objective is not reachable, which often is an unfavorable case in reachability analysis. We have implemented this algorithm in an early prototype and provide some very encouraging experimental results.