For each pair (A, B) of m×n and n×m complex matrices, García-Planas and Sergeichuk (Linear Algebra Appl 302(303):45–61, 1999) constructed a family of pairs (Ags,Bgs) of a simple form to which all matrix pairs (A',B') close to (A, B) can be reduced by transformations (S-1A'R,R-1B'S) given by nonsingular matrices S=S(A',B') and R=R(A',B') that smoothly depend on the entries of A' and B'. We describe all pairs (A, B) of m×n and n×m complex matrices with the following property: all m×m matrices M' close to AB can be reduced to the form AgsBgs by similarity transformations that smoothly depend on the entries of M'.