Organ development involves complex shape transformations driven by active mechanical stresses that sculpt the growing tissue1,2. Epithelial gland morphogenesis is a prominent example where cylindrical branches transform into spherical alveoli during growth3–5. Here we show that this shape transformation is induced by a local change from anisotropic to isotropic tension within the epithelial cell layer of developing human mammary gland organoids. By combining laser ablation with optical force inference and theoretical analysis, we demonstrate that circumferential tension increases at the expense of axial tension through a reorientation of cells that correlates with the onset of persistent collective rotation around the branch axis. This enables the tissue to locally control the onset of a generalized Rayleigh–Plateau instability, leading to spherical tissue buds6. The interplay between cell motion, cell orientation and tissue tension is a generic principle that may turn out to drive shape transformations in other cell tissues.