general conditions for a real, semidefinite matrix to have the eigenvectors associated to highest (real part) eigenvalues be semidefinite

Question

The matrix is not necessarily symmetric. The highest eigenvalue is positive and real (per Frobenius). Assume there is another eigenvalue with real part less than or equal to highest eigenvalue, real part positive, and higher than any other real part of any other eigenvalues. By "semidefinite" eigenvector I mean that the signs of the real part of all non-zero components are equal, since an eigenvector is invariant upon multiplication by a scalar.