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.