  We study avalanches along the hysteresis loop of long-range interacting
spin-glasses with continuous XY-symmetry - which serves as a toy model of
granular superconductors with long-range and frustrated Josephson couplings. We
identify sudden jumps in the $T=0$ configurations of the XY-phases, as an
external field is increased. They are initiated by the softest mode of the
inverse susceptibility matrix becoming unstable, which induces an avalanche of
phase updates (or spin alignments). We analyze the statistics of these events,
and study the correlation between the non-linear avalanches and the soft mode
that initiates them. We find that the avalanches follow the directions of a
small fraction of the softest modes of the inverse susceptibility matrix,
similarly as was found in avalanches in jammed systems. In contrast to the
similar Ising spin-glass (Sherrington-Kirkpatrick) studied previously, we find
that avalanches are not distributed with a scale-free power law, but rather
have a typical size which scales with the system size. We also observe that the
Hessians of the spin-glass minima are not part of standard random matrix
ensembles as the lowest eigenvector has a fractal support.
