  In this paper, the inverse scattering transform for the integrable discrete
nonlocal PT symmetric nonlinear Schr\"odinger equation with nonzero boundary
conditions is presented. According to the two different signs of symmetry
reduction and two different values of the phase difference between plus and
minus infinity, we discuss four cases with significant differences about
analytical regions, symmetry, asymptotic behavior and the presence or absence
of discrete eigenvalues, namely, the existence or absence of soliton solutions.
Therefore, in all cases, we study the direct scattering and inverse scattering
problem, separately. The Riemann-Hilbert problem is constructed and solved as
well as the reconstruction formula of potential is derived, respectively.
Finally, combining the time evolution, we provide the dark, bright, dark-bright
soliton solutions on the nonzero background in difference cases under the
reflectionless condition.
