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Coupling photons to Rydberg excitations in a cold atomic gas yields unprecedentedly large optical nonlinearities at the level of individual light quanta. Here, the basic mechanism exploits the strong interactions between Rydberg atoms to block the formation of nearby dark-state polaritons. However, the dissipation associated with this mechanism ultimately limits the performance of many practical applications. In this work, we propose a new approach to strong photon interactions via a largely coherent mechanism at drastically suppressed photon losses. Rather than a polariton blockade, it is based on an interaction-induced conversion between distinct types of dark-state polaritons with different propagation characteristics. We outline a specific implementation of this approach and show that it permits us to turn a single photon into an effective mirror with a robust and continuously tunable reflection phase. We describe potential applications, including a detailed discussion of achievable operational fidelities.