  The present paper deals with the long-time asymptotic analysis of the initial
value problem for the integrable defocusing nonlocal nonlinear Schr\"odinger
equation $ iq_{t}(x,t)+q_{xx}(x,t)-2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with a
step-like initial data: $q(x,0)\to 0$ as $x\to -\infty$ and $q(x,0)\to A$ as
$x\to +\infty$. Since the equation is not translation invariant, the solution
of this problem is sensitive to shifts of the initial data. We consider a
family of problems, parametrized by $R>0$, with the initial data that can be
viewed as perturbations of the "shifted step function" $q_{R,A}(x)$:
$q_{R,A}(x)=0$ for $x<R$ and $q_{R,A}(x)=A$ for $x>R$, where $A>0$ and $R>0$
are arbitrary constants. We show that the asymptotics is qualitatively
different in sectors of the $(x,t)$ plane, the number of which depends on the
relationship between $A$ and $R$: for a fixed $A$, the bigger $R$, the larger
number of sectors. Moreover, the sectors can be collected into 2 alternate
groups: in the sectors of the first group, the solution decays to 0 while in
the sectors of the second group, the solution approaches a constant (varying
with the direction $x/t=const$).
