pr.probability – Example(s) where replacing a multivariate, discrete RV with a single, univariate RV fail

Let $X_1,ldots,X_n,Y,Z$ be $n+2$ binary random variables and define $X=(X_1,ldots,X_n)$. In most problems, instead of treating $X$ as $n$ distinct binary random variables, there is no loss of generality in treating $X$ as a single variable $U$ that takes on $2^n$ states with the same probabilities (see below for a more rigourous interpretation). For example, $Yperp Z|X iff Yperp Z|U$, and quantities such as entropy and mutual information remain unchanged.

My question: Are there any examples where this replacement “fails”? That is, some property that holds for $(X,Y,Z)$ but doesn’t hold mutatis mutandis for $(U,Y,Z)$?

What I mean by “treating $X$ as a single variable $U$“:
More formally, let $sigma:{1,ldots, 2^n}to {0,1}^n$ be a bijection and enumerate the $2^n$ possible states of $X$ by $sigma$. We can define $U$ to be a random variable on $2^n$ states such that
P(U=k) = P(X=sigma(k)).