  Given a suitable solution $V(t,x)$ to the Korteweg--de Vries equation on the
real line, we prove global well-posedness for initial data $u(0,x) \in V(0,x) +
H^{-1}(\mathbb{R})$.
  Our conditions on $V$ do include regularity but do not impose any assumptions
on spatial asymptotics. We show that periodic profiles $V(0,x)$ $\in
H^5(\mathbb{R}/\mathbb{Z})$ satisfy our hypotheses. In particular, we can treat
localized perturbations of the much-studied periodic traveling wave solutions
(cnoidal waves) of KdV. In our companion paper we show that smooth step-like
initial data also satisfy our hypotheses.
  We employ the method of commuting flows introduced by Killip and Vi\c{s}an;
in the special case $V\equiv 0$, we recover their sharp $H^{-1}(\mathbb{R})$
result.
