Say that two partitions $(P_i)_{i\in I}, (Q_j)_{j\in J}$ are isomorphic iff there is a bijection $f: I\rightarrow J$ such that $\vert P_i\vert=\vert Q_{f(i)}\vert$ for all $i\in I$. (Note that in the absence of choice this is a bit weaker than one might expect.)
Suppose $A$ is a set which supports a tournament (= $[A]^2$ has a choice function). Letting $X$ and $Y$ be the sets of isomorphism types of partitions of $A$ into pairs-or-singletons and at-most-two-pieces respectively, we get an injection $i:X\rightarrow Y$ by picking the "winning" element of each pair. By contrast, a tournament on $A$ (even a linear order on $A$) does not seem to be enough to get the converse situation. I'm interested in the following (more precise) question, essentially as a test of the guess in the previous sentence:
Working in $\mathsf{ZF}$, suppose $A$ is linearly orderable and let $\mathbb{P}$ be the forcing consisting of disjoint pairs of finite subsets of $A$, ordered by $(B,C)\le (B',C')$ iff $B\supseteq B'$ and $C\supseteq C'$. Is it consistent that forcing with $\mathbb{P}$ does add a new isomorphism type of a partition of $A$ into two pieces but does not add a new isomorphism type of a partition of $A$ into pairs-or-singletons?
Precisely, I'm interested in the situation where $V[G]$ (with $G$ $\mathbb{P}$-generic) has a partition of $A$ into two pieces which $V[G]$ thinks is not isomorphic to any partition from $V$, but every partition of $A$ into pairs-or-singletons in $V[G]$ is isomorphic in $V[G]$ to some paritition from $V$. This question is related in spirit, but (I think) not in substance, to a couple earlier questions of mine such as this one.
