  In this paper, we study both the direct and inverse random source problems
associated with the multi-term time-fractional diffusion-wave equation driven
by a fractional Brownian motion. Regarding the direct problem, the
well-posedness is established and the regularity of the solution is
characterized for the equation. In the context of the inverse problem, the
uniqueness and instability are investigated on the determination of the random
source. Furthermore, a reconstruction formula is provided for the phaseless
Fourier modes of the diffusion coefficient in the random source, based on the
variance of the boundary data. To reconstruct the time-dependent source
function from its phaseless Fourier modes, the PhaseLift method, combined with
a spectral cut-off regularization technique, is employed to tackle the phase
retrieval problem. The effectiveness of the proposed method is demonstrated
through a series of numerical experiments.
