Abstract: The emerging generation of radio-interferometric (RI) arrays are set to form images of the sky with a new regime of sensitivity and resolution. This implies a significant increase in visibility data volumes, scaling as \(\mathcal{O}(Q^{2}B)\) for \(Q\) antennas and \(B\) short-time integration intervals (or batches), calling for efficient data dimensionality reduction techniques. This paper proposes a new approach to data compression during acquisition, coined modulated rank-one projection (MROP). MROP compresses the \(Q\times Q\) batchwise covariance matrix into a smaller number \(P\) of random rank-one projections and compresses across time by trading \(B\) for a smaller number \(M\) of random modulations of the ROP measurement vectors. Firstly, we introduce a dual perspective on the MROP acquisition, which can either be understood as random beamforming, or as a post-correlation compression. Secondly, we analyse the noise statistics of MROPs and demonstrate that the random projections induce a uniform noise level across measurements independently of the visibility-weighting scheme used. Thirdly, we propose a detailed analysis of the memory and computational cost requirements across the data acquisition and image reconstruction stages, with comparison to state-of-the-art dimensionality reduction approaches. Finally, the MROP model is validated in simulation for monochromatic intensity imaging, with comparison to the classical and baseline-dependent averaging (BDA) models, and using the uSARA optimisation algorithm for image formation. An extensive experimental setup is considered, with ground-truth images containing diffuse and faint emission and spanning a wide variety of dynamic ranges, and for a range of $uv$-coverages corresponding to VLA and MeerKAT observation.