  We consider a matrix refactorization problem, i.e., a "Lax representation",
for the Yang-Baxter map that originated as the map of polarizations from the
"pure" 2-soliton solution of a matrix KP equation. Using the Lax matrix and its
inverse, a related refactorization problem determines another map, which is not
a solution of the Yang-Baxter equation, but satisfies a mixed version of the
Yang-Baxter equation together with the Yang-Baxter map. Such maps have been
called "entwining Yang-Baxter maps" in recent work. In fact, the map of
polarizations obtained from a pure 2-soliton solution of a matrix KP equation,
and already for the matrix KdV reduction, is NOT in general a Yang-Baxter map,
but it is described by one of the two maps or their inverses. We clarify why
the weaker version of the Yang-Baxter equation holds, by exploring the pure
3-soliton solution in the "tropical limit", where the 3-soliton interaction
decomposes into 2-soliton interactions. Here this is elaborated for pure
soliton solutions, generated via a binary Darboux transformation, of matrix
generalizations of the two-dimensional Toda lattice equation, where we meet the
same entwining Yang-Baxter maps as in the KP case, indicating a kind of
universality.
