The apparent randomness of chaotic eigenstates in interacting quantum systems hides subtle correlations dynamically imposed by their finite energy per particle. These correlations are revealed when Berry’s approach for chaotic eigenfunctions in single-particle systems is lifted into many-body space. We achieve this by a many-body semiclassical analysis, appropriate for the mesoscopic regime of a large but finite number of particles. We thereby identify as signatures of chaotic many-body eigenstates the universality of both their cross-correlations and the Gaussian distribution of expansion coefficients. Combined, these two features imprint characteristic features to the morphology of eigenstates that we check against extensive quantum simulations. The universality of eigenstate correlations for fixed energy density is hence a further signature of many-body quantum chaos that, while consistent with the eigenstate thermalization hypothesis, lies beyond random matrix theory.