According to a popular distinction proposed by the philosopher Ned Block in 1995, our conscious experience would overflow the very limited set of what we can consciously report to ourselves and to others. He proposed to coin this limited consciousness 'Access Consciousness' (A-Cs) and to define 'Phenomenal Consciousness' as a much richer subjective experience that is not accessed but that would still delineate the extent of consciousness. In this article, I review and develop five major problems raised by this theory, and show how a strict A-Cs theory can account for our conscious experience. I illustrate such an A-Cs account within the global workspace (GW) theoretical framework, and revisit some seminal empirical findings and neuropsychological syndromes. In this strict A-Cs perspective, subjective reports are not conceived as the mere passive broadcasting of information to the GW, but as resulting from a dynamic and active chain of internal processes that notably include interpretative and belief attribution stages. Finally, I list a set of testable predictions, of unsolved questions and of some counterintuitive hypotheses.This article is part of the theme issue 'Perceptual consciousness and cognitive access'.