  Given smooth step-like initial data $V(0,x)$ on the real line, we show that
the Korteweg--de Vries equation is globally well-posed for initial data $u(0,x)
\in V(0,x) + H^{-1}(\mathbb{R})$. The proof uses our general well-posedness
result for exotic spatial asymptotics. As a prerequisite, we show that KdV is
globally well-posed for $H^3(\mathbb{R})$ perturbations of step-like initial
data. In the case $V \equiv 0$, we obtain a new proof of the Bona--Smith
theorem using the low-regularity methods that established the sharp
well-posedness of KdV in $H^{-1}$.
