  We compute the spin structure factor of XXZ spin chains in the Heisenberg and
gapped (Ising) regimes in the high-temperature limit for nonzero magnetization,
within the framework of generalized hydrodynamics including diffusive
corrections. The structure factor shows a hierarchy of timescales in the gapped
phase, owing to $s$-spin magnon bound states (`strings') of various sizes.
Although short strings move ballistically, long strings move primarily
diffusively as a result of their collisions with short strings. The interplay
between these effects gives rise to anomalous power-law decay of the spin
structure factor, with continuously varying exponents, at any fixed separation
in the late-time limit. We elucidate the crossover to diffusion (in the gapped
phase) and to superdiffusion (at the isotropic point) in the half-filling
limit. We verify our results via extensive matrix product operator
calculations.
