An electronic device that communicates with a second electronic device is described. During operation, an electronic device communicates first messages with the second electronic device in time slots corresponding to first channels in a first band of frequencies using a peer-to-peer communication protocol (such as neighbor awareness networking or NaN), where, in a given time slot, a given first channel in the first band of frequencies is used in the communication of the first messages. Moreover, the electronic device communicates second messages with the second electronic device in the time slots corresponding to second channels in a second band of frequencies using the peer-to-peer communication protocol, where, in the given time slot, a given second channel in the second band of frequencies is used in the communication of the second messages. Note that the communicating of the first messages and the second messages may at least partially overlap in time.