  We present a rigorous theory of a unified and simple inverse scattering
transform (IST) for both focusing and defocusing real nonlocal
(reverse-space-time) modified Korteweg-de Vries (mKdV) equations with non-zero
boundary conditions (NZBCs) at infinity. The IST problems for the nonlocal
equations with NZBCs are more complicated then ones for the local equations
with NZBCs. The suitable uniformization variable is introduced in order to make
the direct and inverse problems be established on a complex plane instead of a
two-sheeted Riemann surface. The direct scattering problem establishes the
analyticity, symmetries, and asymptotic behaviors of Jost solutions and
scattering matrix, and properties of discrete spectra. The inverse problem is
formulated and solved by means of a matrix-valued Riemann-Hilbert problem. The
reconstruction formula, trace formulae, and theta conditions are obtained.
Finally, the dynamical behaviors of solitons for four different cases for the
reflectionless potentials for both focusing and defocusing nonlocal mKdV
equations with NZBCs are analyzed in detail.
