  We extend the Riemann-Hilbert (RH) method to study the inverse scattering
transformation and high-order pole solutions of the focusing and defocusing
nonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations with
nonzero boundary conditions (NZBCs) at infinity and successfully find its
multiple soliton solutions with one high-order pole and multiple high-order
poles. By introducing the generalized residue formula, we overcome the
difficulty caused by calculating the residue conditions corresponding to the
higher-order poles. In accordance with the Laurent series of reflection
coefficient and oscillation term, the determinant formula of the high-order
pole solution with NZBCs is established. Finally, combined with specific
parameters, the dynamic propagation behaviors of the high-order pole solutions
are further analyzed and some very interesting phenomena are obtained,
including kink solution, anti kink solution, rational solution and
breathing-soliton solution.
