Existing analytical studies of fluid antenna multiple access (FAMA) are largely confined to Rayleigh fading, independent-port assumptions, or asymptotic selection metrics, leaving the fundamental outage behavior under realistic fading and spatial correlation conditions uncharacterized. This paper fills this gap by presenting the first exact and asymptotically tight outage analysis of interference-limited FAMA systems under spatially correlated Nakagami-$m$ fading. By explicitly modeling the joint statistics of the desired and interference powers across fluid-antenna ports, \textcolor{black}{we first formulate the outage probability in terms of the joint desired-signal and interference distributions and then derive an explicit integral expression under a tractable single-reference-port spatial correlation model.} Building on this result, we establish a rigorous diversity-order characterization revealing that the achievable diversity is jointly dictated by the fading severity and the spatial correlation rank imposed by the finite antenna aperture, rather than by the nominal number of ports. To enable accurate and tractable performance evaluation beyond the asymptotic regime, we further introduce a Gamma-matched outage approximation that preserves both the exact diversity order and the array-gain constant. The resulting closed-form expression provides an accurate characterization over a wide range of signal-to-interference ratio (SIR) thresholds and offers clear physical insight into the role of spatial correlation in FAMA systems. Extensive Monte Carlo (MC) simulations confirm the tightness of the exact analysis, the sharpness of the asymptotic results, and the accuracy of the proposed approximation.

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