On matrices having equal corresponding principal minors

Let A, B be n × n matrices with entries in a field F. We say A and B satisfy property $\text{D}$ if B or B^t is diagonally similar to A. It is clear that if A and B satisfy property $\text{D}$, then they have equal corresponding principal minors, of all orders. The question is to what extent the converse is true. There are examples which show the converse is not always true. We modify the problem slightly and give conditions on a matrix A which guarantee that if B is any matrix which has the same principal minors as A, then A and B will satisfy property $\text{D}$. These conditions on A are formulated in terms of ranks of certain submatrices of A and the concept of irreducibility.