We consider the inclusion of a static antiwindup component in a continuous-time low-gain integral controller in feedback with a multi-input multi-output stable linear system subject to an input nonlinearity (from a class of functions that includes componentwise diagonal saturation). We demonstrate that the output of the closed-loop system asymptotically tracks every constant reference vector, which is "feasible" in a natural sense, provided that the integrator gain is sufficiently small. Robustness properties of the proposed control scheme are investigated and three examples are discussed in detail.