The multiparameter eigenvalue problems that arise from the discretization of the multiparameter Sturm-Liouville problems with definiteness constraint possess an inherent structure, which guarantees that all eigenvalues are real. In this paper, we construct a real homotopy method, avoiding the use of complex operations, to find all solutions to this problem. Under a slightly stricter condition, we prove that all eigenvalues can be ordered. Numerical results demonstrate that our homotopy methods are more efficient than existing methods for large-scale problems and can be used to find a specified eigenvalue by tracing only one path.