We first review experimental research on the influence of convection on rod eutectic microstructure, including solidification in space, mechanical stirring, use of a magnetic field, and solidification at high gravity. Then, proposed mechanisms are discussed and their predictions compared with experimental results. It is concluded that vigorous mechanical stirring coarsens the microstructure by altering the concentration field in front of the freezing interface. Only one proposed mechanism is able to explain the observed influence of microgravity on the microstructure of fibrous eutectics. Buoyancy-driven convection was relatively gentle during the directional solidification experiments on earth. We conclude that gentle convection alters the microstructure of a fibrous eutectic only when it causes a fluctuating freezing rate and the kinetics of fiber branching differs from that for fiber termination. These fluctuations may cause the microstructure either to coarsen or to become finer, depending on the relative kinetics of fiber branching and termination. Microgravity experiments and theoretical modeling are proposed to confirm the proposed mechanism. The microgravity experiments would utilize electric current pulses during solidification in order to induce a gentle oscillation in the freezing rate. This transient process would be modeled theoretically, taking into account interface growth kinetics, the lead of the fibers in from of the matrix, and branching kinetics, without the classical assumption of minimum supercooling.