We pick up a global routing model using rhomboidal tiles due to Hähnle and Saccardi and expand on their ideas. We add continuous via costs to their model and introduce a polyhedral description of rectilinear graphs with a fixed structure. Using this description, we derive a proof that minimum cost Steiner trees, rather than paths, lie on the rhomboidal Hanan grid and extend goal-oriented path search techniques to the rhomboidal model. We prove the competitiveness of this continuous routing approach by comparing subsequent detailed routing results to a traditional global router on the 5 nm technology node.