Morphological shape changes of giant tubular vesicles prepared from the diblock copolymer polybutadiene-(32)-b-polyethylene oxide(20) (PB-PEO) in aqueous solution after thermal quenches between 10 and. 50 K were monitored via quantitative phase-contrast microscopy. Reducing the temperature leads to extremely slow sequential beading of the tubes where the formation of necks starts symmetrically at the two ends. We characterize the neck diameters and find that the necks close one by one with effective velocities on the order of a few tens of nanometers per minute. The necks do not close continuously, but rather their radii decrease in time in a sequence of exponential decays between intermediate plateaus. The slow dynamics is a result of the high membrane surface viscosity of PB-PEO. Sequential beading is rationalized via a cascade of metastable shapes determined by the bending elastic energy of the tubular polymersomes.