Acoustic handling of nanoparticles in resonating acoustofluidic devices is often impeded by the presence of acoustic streaming. For micrometer-sized acoustic chambers, this acoustic streaming is typically driven by viscous shear in the thin acoustic boundary layer near the fluid-solid interface. Alternating current (ac) electroosmosis is another boundary-driven streaming phenomenon routinely used in microfluidic devices for the handling of particle suspensions in electrolytes. Here, we study how streaming can be suppressed by combining ultrasound acoustics and ac electroosmosis. Based on a theoretical analysis of the electrokinetic problem, we are able to compute numerically a form of the electrical potential at the fluid-solid interface, which is suitable for suppressing the typical acoustic streaming pattern associated with a standing acoustic half-wave. In the linear regime, we even derive an analytical expression for the electroosmotic slip velocity at the fluid-solid interface and use this as a guiding principle for developing models in the experimentally more relevant nonlinear regime that occurs at elevated driving voltages. We present simulation results for an acoustofluidic device, showing how implementing a suitable ac electroosmosis results in a suppression of the resulting electroacoustic streaming in the bulk of the device by 2 orders of magnitude.