  We study the inverse random source problem for the time-space fractional
diffusion equation driven by fractional Brownian motion with Hurst index
$H\in(0,1)$. With the aid of a novel estimate, by using the operator approach
we propose regularity analyses for the direct problem. Then we provide a
reconstruction scheme for the source terms $f$ and $g$ up to the sign. Next,
combining the properties of Mittag-Leffler function, the complete uniqueness
and instability analyses are provided. It's worth mentioning that all the
analyses are unified for $H\in(0,1)$.
