In this report, we consider a kind of multi-term Caputo tempered fractional stochastic differential equations and prove the existence and uniqueness of the true solution. Then we derive an Euler-Maruyama (EM) scheme to solve the considered equations. In view of the huge computational cost caused by the EM scheme to achieve reasonable accuracy, a fast EM scheme is proposed based on the sum-of-exponentials approximation to improve its computational efficiency. Moreover, the strong convergence of our two numerical schemes are proved. Finally, several numerical examples are carried out to support our theoretical results and demonstrate the superior computational efficiency of the fast EM scheme.