I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby undermined the entire answer. However, it can be salvaged if there exists a function $\det$ defined on all real-valued matrices (not just the square ones) having the following properties.
- $\det$ is real-valued
- $\det$ has its usual value for square matrices
- $\det(AB)$ always equals $\det(A)\det(B)$ whenever the product $AB$ is defined.
- $\det(A) \neq 0$ iff $\det(A^\top) \neq 0$
Does such a function exist?