Printed circuit substrates may be formed from rigid printed circuit material or flexible sheets of polymer. Printed circuit substrates may have conductive traces. Integrated circuits, touch sensor electrode structure, sensors, and other components may be mounted to the conductive traces. Connectors such as board-to-board connectors may be used to couple printed circuit substrates together. To hold the connectors together and to provide electromagnetic shielding, printed circuits and connectors may be surrounded by printed circuit connector securing structures. The printed circuit connector securing structures may have one or more strips of conductive fabric tape wrapped around the connectors. Metal stiffening members may be attached to opposing ends of the strip of conductive tape to facilitate removal of the tape for rework or repair. An additional strip of tape may be used to help secure the wrapped conductive tape. The additional strip may have a tab to facilitate removal.