Life in complex systems, such as cities and organisms, comes to a standstill when global coordination of mass, energy, and information flows is disrupted. Global coordination is no less important in single cells, especially in large oocytes and newly formed embryos, which commonly use fast fluid flows for dynamic reorganization of their cytoplasm. Here, we combine theory, computing, and imaging to investigate such flows in the Drosophila oocyte, where streaming has been proposed to spontaneously arise from hydrodynamic interactions among cortically anchored microtubules loaded with cargo-carrying molecular motors. We use a fast, accurate, and scalable numerical approach to investigate fluid-structure interactions of 1000s of flexible fibers and demonstrate the robust emergence and evolution of cell-spanning vortices, or twisters. Dominated by a rigid body rotation and secondary toroidal components, these flows are likely involved in rapid mixing and transport of ooplasmic components.