  We investigate the large-time asymptotics of solution for the Cauchy problem
of the nonlocal focusing modified Kortweg-de Vries (MKdV) equation with
step-like initial data, i.e., $u_0(x)\rightarrow 0$ as $x\rightarrow-\infty$,
$u_0(x)\rightarrow A$ as $x\rightarrow+\infty$,where $A$ is an arbitrary
positive real number. We firstly develop the direct scattering theory to
establish the basic Riemann-Hilbert (RH) problem associated with step-like
initial data. Thanks to the symmetries $x\rightarrow-x$, $t\rightarrow-t$ of
nonlocal MKdV equation, we investigate the asymptotics for
$t\rightarrow-\infty$ and $t\rightarrow+\infty$ respectively. Our main
technique is to use the steepest descent analysis to deform the original
matrix-valued RH problem to corresponded regular RH problem, which could be
explicitly solved. Finally we obtain the different large-time asymptotic
behaviors of the solution of the Cauchy problem for focusing nonlocal MKdV
equation in different space-time sectors $\mathcal{R}_{I}$, $\mathcal{R}_{II}$,
$\mathcal{R}_{III}$ and $\mathcal{R}_{IV}$ on the whole $(x,t)$-plane.
