  In this paper, we consider the Cauchy problem for an integrable real nonlocal
(also called reverse-space-time) mKdV equation with nonzero boundary conditions
\begin{align*} &q_t(x,t)-6\sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0,
&q(x,0)=q_{0}(x),\lim_{x\to \pm\infty} q_{0}(x)=q_{\pm}, \end{align*} where
$|q_{\pm}|=1$ and $q_{+}=\delta q_{-}$, $\sigma\delta=-1$. Based on the
spectral analysis of the Lax pair, we express the solution of the Cauchy
problem of the nonlocal mKdV equation in terms of a Riemann-Hilbert problem. In
a fixed space-time solitonic region $-6<x/t<6$, we apply
$\bar{\partial}$-steepest descent method to analyze the long-time asymptotic
behavior of the solution $q(x,t)$. We find that the long time asymptotic
behavior of $q(x,t)$ can be characterized with an $N(\Lambda)$-soliton on
discrete spectrum and leading order term $\mathcal{O}(t^{-1/2})$ on continuous
spectrum up to an residual error order $\mathcal{O}(t^{-1})$.
