The effect in tiger eye is called chatoyancy, and it comes from the internal structure: parallel fibres of a blue mineral that were partly dissolved and replaced by silica, leaving them aligned. Light hits the fibres and is scattered sideways rather than reflected straight back, which is what makes the band move.
Because the fibres run in one direction, the beads have to be cut to face them, and on a good strand every bead shows the band. Natural stone, not dyed glass — the giveaway for glass is the absence of the moving band at any angle.