We report on a study of the kinetics of alignment and decay of alignment, using small-angle neutron scattering (SANS), of a complex fluid under Couette shear containing a highly entangled network of threadlike micelles. The time scales of alignment revealed are dominated by the collective properties of the network rather than the individual micellar segments. This results in rates orders of magnitude slower than any previously observed in similar systems as well as in the first observation of an alignment proceeding in two stages.