A graph is chordal if every cycle of length at least four contains a chord, that is, an edge connecting two nonconsecutive vertices of the cycle. Several classical applications in sparse linear systems, database management, computer vision, and semidefinite programming can be reduced to finding the minimum number of edges to add to a graph so that it becomes chordal, known as the minimum chordal completion problem (MCCP). We propose a new formulation for the MCCP that does not rely on finding perfect elimination orderings of the graph, as has been considered in previous work. We introduce several families of facet-defining inequalities for cycle subgraphs and investigate the underlying separation problems, showing that some key inequalities are NP-Hard to separate. We also identify conditions through which facets and inequalities associated with the polytope of a certain graph can be adapted in order to become facet defining for some of its subgraphs or supergraphs. Numerical studies combining heuristic separation methods and lazy-constraint generation indicate that our approach substantially outperforms existing methods for the MCCP.