When do the r-by-r minors of a matrix form a tropical basis?

We show that the r-by-r minors of a d-by-n matrix of variables form a tropical basis of the ideal they generate if and only if one of the following conditions holds: (i) r⩽3, (ii) r=min{d,n}, (iii) r=4 and min{d,n}⩽6. This answers a question raised by M. Chan, A. Jensen, and E. Rubei.