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Knowledge of the three-dimensional structure of a charged particle beam bunch is essential for understanding its evolution and for initializing computer simulations, especially when space charge is involved. This paper presents a novel experimental method for time-sliced mapping of the transverse phase space of a space-charge dominated beam based on tomographic principles. The combination of a high precision tomographic diagnostic with fast imaging screens and a gated camera are used to produce phase-space maps of two beams: one with a parabolic current profile and another with a short perturbation atop a rectangular pulse. The correlations between longitudinal and transverse phase spaces are apparent and their impact on the dynamics is discussed.