  We initiate the study of positive-tropical generators as positive analogues
of the concept of tropical bases. Applying this to the tropicalization of
determinantal varieties, we develop criteria for characterizing their positive
part. We focus on the study of low-rank matrices, in particular matrices of
rank 2 and 3. Moreover, in the case square-matrices of corank 1, we fully
classify the signed tropicalization of the determinantal variety, even beyond
the positive part.
