We find novel perturbative fixed points by introducing mildly spacetime dependent couplings into otherwise marginal terms. In four-dimensional quantum field theory, these are physical analogues of the small-$ϵ$ Wilson-Fisher fixed point. Rather than considering $4\ensuremath{-}ϵ$ dimensions, we stay in four dimensions but introduce couplings whose leading spacetime dependence is of the form $\ensuremath{\lambda}{x}^{\ensuremath{\kappa}}{\ensuremath{\mu}}^{\ensuremath{\kappa}}$, with a small parameter $\ensuremath{\kappa}$ playing a role analogous to $ϵ$. We show, in ${\ensuremath{\phi}}^{4}$ theory and in QED and QCD with massless flavors, that this leads to a critical theory under perturbative control over an exponentially wide window of spacetime positions $x$. The exact fixed point coupling ${\ensuremath{\lambda}}_{*}(x)$ in our theory is identical to the running coupling of the translationally invariant theory, with the scale replaced by $1/x$. Similar statements hold for three-dimensional ${\ensuremath{\phi}}^{6}$ theories and two-dimensional sigma models with curved target spaces. We also describe strongly coupled examples using conformal perturbation theory.