Entanglement is at the heart of quantum physics, both for its conceptual foundations and for applications in quantum communication. Remarkably, entanglement can be ‘swapped': if we prepare two independent entangled pairs A1–A2 and B1–B2, a joint measurement on A1 and B1 (called a ‘Bellstate measurement', BSM) has the effect of projecting A2 and B2 onto an entangled state, although these two particles have never interacted nor share any common past1,2. Entanglement swapping with photon pairs has already been experimentally demonstrated3–6 using pulsed sources—where the challenge was to achieve sufficiently sharp synchronization of the photons in the BSM—but never with continuous-wave sources, as originally proposed2. Here, we present an experiment where the coherence time of the photons exceeds the temporal resolution of the detectors.Hence, photon timing can be obtained by the detection times, and pulsed sources can be replaced by continuous-wave sources, which do not require any synchronization6,7. This allows for the first time the use of completely autonomous sources, an important step towards real-world quantum networks with truly independent and distant nodes.