A complete map M_n consists of n disk-like regions, where every “country” shares a non-zero-length border-line with all the other n-1 countries. Different methods are described for constructing easy-to-understand visualization models of complete maps. One useful representation takes the form of a 2-layered “pillow-case” forming a disk with a minimum number of tunnels. Results are presented for complete maps with eight to twelve countries on symmetrical handle-bodies of genus-2 through genus-6.