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Synchronized spin-valve oscillators may lead to nanosized microwave generators that do not require discrete elements such as capacitors or inductors. Uniformly magnetized oscillators have been synchronized, but offer low power. Gyrating magnetic vortices offer greater power, but vortex synchronization has yet to be demonstrated. Here we find that vortices can interact with each other through the mediation of antivortices, leading to synchronization when they are closely spaced. The synchronization does not require a magnetic field, making the system attractive for electronic device integration. Also, because each vortex is a topological soliton, this work presents a model experimental system for the study of interacting solitons.