<p>This paper explores the nonlocal sine-Gordon equation with non-vanishing boundary conditions, this equation displays drastically different symmetry properties compared to its local counterpart, resulting in a disparate discrete spectral distribution. For getting <i>N</i>-simple and <i>N</i>-double poles soliton solutions of the nonlocal sine-Gordon equation, we adopt inverse scattering transform based on Riemann-Hilbert problem. First, we define eigenfunctions from spatial dependent scattering problem, analyze their analytical properties, symmetry relations and asymptotic behaviors. Then we considering the time evolution of the scattering data. The next we construct and solve the corresponding Riemann-Hilbert problem. Finally, we display dynamical behaviors of these solutions graphically with 3D and projection profiles, such as the anti-dark soliton, <i>M</i>-type solitons, single-peaked breather, double-peaked breather, Akhmediev breather and Kuznetsov-Ma breather. These solutions play a significant role in revealing the abundant dynamics of solitons and advancing our comprehension of nonlocal nonlinear phenomena.</p>