In spatiotemporal systems with advection, suppression of noise-sustained structures involves questions that are outside of the framework of deterministic dynamical systems control (such as Ott-Grebogi-Yorke-type methods). Here we propose and test an alternate strategy where a nonlocal additive feedback is applied, with the objective to create a new deterministic solution that becomes robust to noise. As a remarkable fact-though the needed parameter perturbations required have essentially a finite size-they turn out to be extraordinarily small in principle: 10;{-8} in the free-electron laser experiment presented here.