We find novel perturbative fixed points by introducing mildly spacetime-dependent couplings into otherwise marginal terms. In four-dimensional QFT, these are physical analogues of the small-$\epsilon$ Wilson-Fisher fixed point. Rather than considering $4-\epsilon$ dimensions, we stay in four dimensions but introduce couplings whose leading spacetime dependence is of the form $\lambda x^\kappa \mu^\kappa$, with a small parameter $\kappa$ playing a role analogous to $\epsilon$. We show, in $\phi^4$ 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 $\lambda_*(x)$ 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 $\phi^6$ theories and two-dimensional sigma models with curved target spaces. We also describe strongly coupled examples using conformal perturbation theory. Show Partial Abstract