Mechanical interactions play a crucial role in epithelial morphogenesis, yet understanding the complex mechanisms through which stress and deformation affect cell behaviour remains an open problem. Here we formulate and analyse the active tension network (ATN) model, which assumes that the mechanical balance of cells within a tissue is dominated by cortical tension and introduces tension-dependent active remodelling of the cortex. We find that ATNs exhibit unusual mechanical properties. Specifically, an ATN behaves as a fluid at short times, but at long times supports external tension like a solid. Furthermore, an ATN has an extensively degenerate equilibrium mechanical state associated with a discrete conformal—‘isogonal’—deformation of cells. The ATN model predicts a constraint on equilibrium cell geometries, which we demonstrate to approximately hold in certain epithelial tissues. We further show that isogonal modes are observed in the fruit fly embryo, accounting for the striking variability of apical areas of ventral cells and helping understand the early phase of gastrulation. Living matter realizes new and exotic mechanical states, the study of which helps to understand biological phenomena.