<p>In this paper, the inverse scattering transform for the integrable inhomogeneous reverse time nonlocal lattice with nonzero boundary condition is presented. According to the different signs of reverse-time reduction, we discuss two lattices with significant differences about analytical regions, symmetry, asymptotic behavior and discrete eigenvalues in terms of a suitable uniform variable. Therefore, we study the direct scattering problem, separately. For the inverse scattering problem, the Riemann-Hilbert problem is constructed and solved as well as the reconstruction formula of potential is derived together. Finally, combining the time evolution, we provide the breather, oscillating soliton, period and dark-bright soliton solutions on the nonzero background in different cases under the reflectionless condition.</p>