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Let's say $X, Y$ are dependent random variables. Can I find an example such that for a transcript $t$ conditioned on a communication protocol $\Pi=t$, the variables become independent?

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Yes. Consider the following protocol. Alice flips a coin, and sets $X$ and $Y$ to be equal to the outcome of the coin flip. She then sends $X,Y$ to Bob. Note that the transcript will include the value of $X,Y$.

Now unconditionally, $X,Y$ are dependent (there is probability $1/2$ that they are Heads,Heads and probability $1/2$ that they are Tails,Tails). But conditioned on the transcript, $X,Y$ are independent (e.g., if the transcript says they are Heads, then with conditional probability $1$ they are Heads,Heads and all other cases have conditional probability $0$; that meets the definition for independence).

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