Example: Equal records, different residuals

For a qubit input and $z\in \{0,1\}$, define

Both are complete instruments with effects $|z⟩⟨z|$. Thus they have identical Born record laws for every input. Their conditional quantum outputs differ on every positive branch.

Both instruments are represented by Enter followed by allocation and, for $\Psi$, a fixed $X$ gate on the fresh output. They do not define two different Enter strategies. A later computational readout of the returned output distinguishes them perfectly.