  We study the long time asymptotic behavior for the Cauchy problem of the
modified Camassa-Holm (mCH) equation with step-like initial data \begin{align}
&m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}=0, \quad m=u-u_{xx},
\nonumber \\ &u(x,0)=u_0(x)\to \left\{ \begin{array}{ll}
  A_1, &\ x\to+\infty,\\[5pt]
  A_2, &\ x\to-\infty,
  \end{array}\right.\nonumber \end{align} where $A_1$ and $A_2$ are two
positive constants. Our main technical tool is the representation of the Cauchy
problem with an associated matrix Riemann-Hilbert (RH) problem and the
consequent asymptotic analysis of this RH problem. Based on the spectral
analysis of the Lax pair associated with the mCH equation and scattering
matrix, the solution of the step-like initial problem is characterized via the
solution of a RH problem in the new scale $(y,t)$. We adopt double coordinates
$(\xi, c)$ to divide the half-plane $\{ (\xi,c): \xi \in \mathbb{R}, \ c> 0, \
\xi=y/t\}$ into four asymptotic regions. Further using the Deift-Zhou steepest
descent method, we derive different long time asymptotic expansion of the
solution $u(y,t)$ in different space-time regions by the different choice of
g-function. The corresponding leading asymptotic approximations are given with
the slow/fast decay step-like background wave in genus-0 regions and elliptic
waves in genus-2 regions. The second term of the asymptotics is characterized
by Airy function or parabolic cylinder model. Their residual error order is
$\mathcal{O}(t^{-1})$ or $\mathcal{O}(t^{-2})$ respectively.
