During cytokinesis, a contractile actomyosin ring constricts and divides the cell in two. How the ring marshals actomyosin forces to generate tension is not settled. Recently, a superresolution microscopy study of the fission yeast ring revealed that myosins and formins that nucleate actin filaments colocalize in plasma membrane-anchored complexes called nodes in the constricting ring. The nodes move bidirectionally around the ring. Here we construct and analyze a coarse-grained mathematical model of the fission yeast ring to explore essential consequences of the recently discovered ring ultrastructure. The model reproduces experimentally measured values of ring tension, explains why nodes move bidirectionally, and shows that tension is generated by myosin pulling on barbed-end-anchored actin filaments in a stochastic sliding-filament mechanism. This mechanism is not based on an ordered sarcomeric organization. We show that the ring is vulnerable to intrinsic contractile instabilities, and protection from these instabilities and organizational homeostasis require both component turnover and anchoring of components to the plasma membrane.