Communication in a network generally takes place through a sequence of intermediate nodes connected by communication channels. In the standard theory of communication, it is assumed that the communication network is embedded in a classical spacetime, where the relative order of different nodes is well defined. In principle, a quantum theory of spacetime could allow the order of the intermediate points between sender and receiver to be in a coherent superposition. Here we experimentally realize a tabletop simulation of this exotic possibility on a photonic system, demonstrating high-fidelity transmission of quantum information over two noisy channels arranged in a superposition of two alternative causal orders.