Recent years have seen a growing interest in developing coherent atom-light interfaces due to their relevance for cavity QED and quantum networks. The focus has been on systems with collectively coupled ensembles, and with strongly coupled single atoms. However, combining strong atom-light coupling with single-atom control remains challenging. We report on experiments with an atomic tweezer array that is strongly coupled to an optical fiber cavity. The setup integrates three ingredients: single atom control for arbitrary quantum logical operations, a tweezer-cavity system for cavity QED, and a direct fiber interface for real-time measurements and networking. This opens a path for programmable interactions within an atomic tweezer array, for non-destructive readout protocols, and for implementing quantum network protocols.