Abstract: Pair-density waves (PDWs) are unconventional superconductors in which the order parameter periodically oscillates in space. Compared to other superconductors, PDWs are particularly susceptible to disorder, since random impurities can disrupt the periodic structure that defines them. However, it is not clear whether disorder completely destroys the PDW or instead gives rise to a new phase. Here, we show that a new phase can indeed arise, by considering a strongly inhomogeneous limit in which the system consists of a random collection of PDW puddles embedded in a metallic background*. When the puddles are dilute, they become phase coherent at low temperatures, resulting in a state that is macroscopically equivalent to an s-wave superconductor. This can be viewed as an example of order-parameter fractionalization, in which the PDW order splits into an s-wave superconducting component and a charge-density-wave component, the latter of which is destroyed by disorder. This result highlights the unexpected robustness of superconductivity in PDWs and provides new insights into the effects of disorder on d-wave superconductors.