  We consider the focusing integrable nonlocal nonlinear Schr\"odinger equation
\[\mathrm{i}q_{t}(x,t)+q_{xx}(x,t)+2q^{2}(x,t)\bar{q}(-x,t)=0\] with asymmetric
nonzero boundary conditions: $q(x,t)\to\pm A\mathrm{e}^{-2\mathrm{i}A^2t}$ as
$x\to\pm\infty$, where $A>0$ is an arbitrary constant. The goal of this work is
to study the asymptotics of the solution of the initial value problem for this
equation as $t\to+\infty$. For a class of initial values we show that there
exist three qualitatively different asymptotic zones in the $(x,t)$ plane.
Namely, there are regions where the parameters are modulated (being dependent
on the ratio $x/t$) and a central region, where the parameters are unmodulated.
This asymptotic picture is reminiscent of that for the defocusing classical
nonlinear Schr\"odinger equation, but with some important differences. In
particular, the absolute value of the solution in all three regions depends on
details of the initial data.
