This paper studies a problem frequently seen in wireless networks: Every node wishes to broadcast information to nodes within a single hop, which are referred to as its peers. We call this problem mutual broadcast. A novel solution is proposed, which exploits the multiaccess nature of the wireless medium and addresses the half-duplex constraint at the fundamental level. The defining feature of the scheme is to let all nodes send their messages at the same time, where each node broadcasts a codeword (selected from its unique codebook) consisting of on-slots and off-slots, where it transmits only during its on-slots, and listens to its peers through its own off-slots. Decoding can be viewed as a problem of sparse support recovery based on linear measurements. In case each message consists of a small number of bits, an iterative message-passing algorithm based on belief propagation is developed, the performance of which is characterized using a state evolution formula in the limit where each node has a large number of peers. Numerical results demonstrate that, to achieve the same reliability for mutual broadcast, the proposed scheme achieves three to five times the rate of ALOHA and carrier-sensing multiple-access (CSMA) in typical scenarios.