Reactive programming over a network is a challenging task because efficient elimination of temporary violations of data flow invariants, known as glitches, in a distributed setting is still an open issue. In this paper, we propose a method for constructing a distributed reactive programming system of which runtime guarantees the properties of single source glitch-freedom and the robustness against out-of-order messages. Based on the method, we developed a purely functional reactive programming language XFRP whose compiler produces Erlang code. Using some examples, we show that the proposed method is beneficial for constructing distributed reactive applications without suffering from inconsistencies.