The development of intestinal organoids from single adult intestinal stem cells in vitro recapitulates the regenerative capacity of the intestinal epithelium^,. Here we unravel the mechanisms that orchestrate both organoid formation and the regeneration of intestinal tissue, using an image-based screen to assay an annotated library of compounds. We generate multivariate feature profiles for hundreds of thousands of organoids to quantitatively describe their phenotypic landscape. We then use these phenotypic fingerprints to infer regulatory genetic interactions, establishing a new approach to the mapping of genetic interactions in an emergent system. This allows us to identify genes that regulate cell-fate transitions and maintain the balance between regeneration and homeostasis, unravelling previously unknown roles for several pathways, among them retinoic acid signalling. We then characterize a crucial role for retinoic acid nuclear receptors in controlling exit from the regenerative state and driving enterocyte differentiation. By combining quantitative imaging with RNA sequencing, we show the role of endogenous retinoic acid metabolism in initiating transcriptional programs that guide the cell-fate transitions of intestinal epithelium, and we identify an inhibitor of the retinoid X receptor that improves intestinal regeneration in vivo.