  It is concerned with the large-time asymptotics of the Cauchy problem of the
defocusing modified Korteweg-de Vries (mKdV) equation with step-like initial
data subject to compact perturbations, that is, \begin{align*}
  q_{0}(x)-q_{0c}(x)=0, \ \text{for} \ |x|>N \end{align*} with some positive
$N$, where \begin{align*}
  q_{0c}(x)=\left\{
  \begin{aligned}
  &c_{l}, \quad x\leqslant 0,
  &c_{r}, \quad x>0,
  \end{aligned}
  \right. \end{align*} and $c_l>c_{r}>0$. It follows from the standard direct
and inverse scattering theory that an RH characterization for the step-like
problem is constructed. By performing the nonlinear steepest descent analysis,
we mainly derive the large-time asymptotics in the each of four asymptotic
zones in the $(x,t)$-half plane.
