The canonical mechanism that drives cell division comprises the formation and constriction of a contractile actin ring^{1, 2–3}. However, this mechanism is not compatible with the early development of many vertebrates^{4, 5, 6, 7, 8–9}. Yolk-anchored embryos typically cannot form a complete ring during early cleavage divisions, but it remains unclear how a partial circular arc with loose ends can divide the cell. Here, by combining laser ablation of the cytokinetic band with rheological measurements in vivo, we show that stiffening of the bulk cytoplasm, mediated by the interphase microtubule network, stabilizes the contractile band by anchoring it along its length during growth. Conversely, as the cell cycle progresses, the cytoplasm fluidizes, diminishing band–cytoplasmic anchoring and facilitating band ingression. This dynamic interplay between stability and growth versus instability and ingression repeats for several cell cycles until division is complete, resulting in a mechanical ratchet that drives cell division. Our study underscores the role of temporal control over cytoplasmic rheology as a key feature that drives unilateral cytokinesis in the absence of a closed actin ring.