A variety of real-time multicast applications such as video conferences, remote lectures, and video-on-demand have become in commonplace with the expansion of broadband Internet services. Due to nontrivial problems in the IP multicast technology, the peer-to-peer multicast technology (P2P-multicast) has emerged as a practical implementation, although its network resource utilization is less efficient. A multihome network has the potential of alleviating this inefficiency by providing flexibility in communication path selections for each host with multiple gateways to the Internet. This paper has first formulated the P2P-multicast routing problem in the multihome network, and has proved the NP-completeness of its decision problem. Then, a two-stage heuristic algorithm called P2PMM_router has been presented for this P2P Multicast Multihome-network routing problem. The first stage constructs an initial multicast routing tree from an optimum spanning tree by Prim algorithm, through satisfying the constraints. The second stage improves the tree by repeating partial modifications and constraint satisfactions. The extensive simulation results using random network instances support the effectiveness of our P2PMM_router.