An electronic device may be provided with a dielectric cover and a phased antenna array for conveying millimeter wave signals. A conductive pocket may be mounted to the cover. The pocket may include a conductive rear wall and conductive sidewalls that extend from a periphery of the rear wall to the cover. The array may be mounted to the rear wall and may convey signals through the cover. The sidewalls may extend from the cover at non-zero angles with respect to the normal axis of the cover. The shape of the pocket and the cover may be selected so that the pocket is non-resonant at frequencies handled by the array, to mitigate destructive interference within the pocket, to block surface waves from propagating along the cover, and to tweak the radiation pattern of the array to exhibit a desired shape and directionality.