Epithelial tissue morphodynamics is a crucial aspect in metazoan development that lends itself to be studied by means of in vitro representative models. Cysts serve as a paradigm for the formation of hollow spherical structures in vivo. The orderly development of cell aggregates into spheroidal monolayers delimiting a single lumen is a process subject to mechanical constraints and depends on the establishment and maintenance of cellular apico-basal polarity. In order to characterize in detail the geometry and topology of growing cysts, we reconstructed their three-dimensional structure by means of computer-assisted segmentation. We show that cell-cell contact statistics bears a clear signature of the out-of-equilibrium character of cyst morphogenesis. Out-of-equilibrium dynamics drives the appearance of aberrant multiluminal phenotypes, unless strict control of cell division geometry is enforced. Theoretical modeling and numerical simulations allow for a quantitative comparison with experimental results and unveil the presence of multiple local equilibria associated with geometrically frustrated configurations. The single-lumen phenotype is rescued and the quasi-equilibrium topology is recovered upon inducing tissue fluidization through the inhibition of ROCK activity.