  In this paper, the nonlocal reverse space-time derivative nonlinear
Schr\"odinger equation under nonzero boundary conditions is investigated using
the Riemann-Hilbert (RH) approach. The direct scattering problem focuses on the
analyticity, symmetries, and asymptotic behaviors of the Jost eigenfunctions
and scattering matrix functions, leading to the construction of the
corresponding RH problem. Then, in the inverse scattering problem, the Plemelj
formula is employed to solve the RH problem. So the reconstruction formula,
trace formulae, $\theta$ condition, and exact expression of the single-pole and
double-pole solutions are obtained. Furthermore, dark-dark solitons,
bright-dark solitons, and breather solutions of the reverse space-time
derivative nonlinear Schr\"odinger equation are presented along with their
dynamic behaviors summarized through graphical simulation.
