Is a geometric construction of the dual RSK correspondence along the lines of Viennot's "light and shadows construction" written up somewhere? This is a bijective correspondence between 01 matrices and pairs of SSYT with mutually transpose shapes.
As far as I know, the Viennot's construction only works for RS algorithm taking permutations as input. For matrices there's a generalised version called matrixball construction, which can be found in 4.2 of "Young Tableaux" by William Fulton. I am not sure whether there is a matrixball construction for dual RSK algorithm.
However, if only consider permutations, then by replacing "row" to "column" in the description of RSK algorithm we get dual RSK algorithm. That means, the two tableaux obtained by dual RSK are exactly transposes of the pair by RSK, from the same permutation. Therefore the "light and shadow" construction still works for dual RSK and all one needs to do is place the readings to the columns rather than rows.
Edit: There is a matrixball construction for socalled Burge correspondence, see appendix A.4.1 of Fulton's book. Burge correspondence allows a nonnegative integer matrix input for column insertion algorithm (by reading the matrix entries from right to left and top to bottom). This covers the case of words and permutations where each row only has one nonzero entry. I don't know the 01 matrix case yet. However, the description of matrixball construction is a bit too concise here for me to understand, especially how a new matrix of balls is constructed from an old one.

$\begingroup$ True, the matrix ball construction generalizes the RS algorithm to RSK, but it seems no similar generalization is available for the dual RSK. $\endgroup$ – Amritanshu Prasad Feb 11 '13 at 5:37
It's all written up rather nicely in Heather Dornom's Honours thesis from 2005. She gives a version of the matrix ball construction that works in these cases and also explains growth models for the RSK correspondence and its dual.