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The disruptive reconfigurable intelligent surface (RIS) technology is steadily gaining relevance as a key element in future 6G networks. However, a one-size-fits-all RIS hardware design is yet to be defined due to many practical considerations. A major roadblock for currently available RISs is their inability to concurrently operate at multiple carrier frequencies, which would lead to redundant installations to support multiple radio access technologies (RATs). In this paper, we introduce FABRIS, a novel and practical multi-frequency RIS design. FABRIS is able to dynamically operate across different radio frequencies (RFs) by means of frequency-tunable antennas as unit cells with virtually no performance degradation when conventional approaches to RIS design and optimization fail. Remarkably, our design preserves a sufficiently narrow beamwidth as to avoid generating signal leakage in unwanted directions and a sufficiently high antenna efficiency in terms of scattering parameters. Indeed, FABRIS selects the RIS configuration that maximizes the signal at the intended target user equipment (UE) while minimizing leakage to non-intended neighboring UEs. Numerical results and full-wave simulations validate our proposed approach against a naive implementation that does not consider signal leakage resulting from multi-frequency antenna arrays.