And what are it's applications? A conceptual explanation would be great! Is there an expository note about this somewhere?
Some references have already appeared in the answers and comments below. To make the question more specific, classical RSK has combinatorial interpretations in terms of symmetric functions, for example. If RSK gives the Cauchy identity: $$\sum_{\lambda} s _{\lambda}(x)s _{\lambda}(y)=\prod _{i,j} \frac{1}{1-x_iy_j}$$ what is an analogous interpretation for tropical RSK? (From some buzzwords I've heard in a few talks recently, it probably has something to do with shifted or elliptic Schur functions.)