Let $S$ be a nonempty finite state space, let $P:S \times S \to [0,1]$ be a Markov transition matrix, and let $\pi:S \to [0,1]$ be a stationary distribution for $P$, meaning $\sum_{x \in S} \pi(x)=1$ and, for every $y \in S$, $\sum_{x \in S}\pi(x)P(x,y)=\pi(y)$. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and let $(F_n)_{n \leq -1}$ be independent identically distributed measurable random maps $F_n:(\Omega,\mathcal{F}) \to (S^S,2^{S^S})$, where $S^S$ denotes the finite set of all maps $S \to S$, such that, for every $n \leq -1$ and every $x,y \in S$, $\mathbb{P}(\{\omega \in \Omega:F_n(\omega)(x)=y\})=P(x,y)$. For integers $a \leq b \leq 0$, define the reused-randomness composition $\Phi_{a,b}:\Omega \to S^S$ by $\Phi_{a,a}(\omega)=\operatorname{id}_S$ and, when $a<b$, by $\Phi_{a,b}(\omega)=F_{b-1}(\omega)\circ F_{b-2}(\omega)\circ \cdots \circ F_a(\omega)$. Define the coalescence time $\tau:\Omega \to \mathbb{N}\cup\{\infty\}$ by $\tau(\omega)=\inf\{t\in\mathbb{N}:\Phi_{-t,0}(\omega)\text{ is constant on }S\}$, with $\inf\varnothing=\infty$. Assume $\tau<\infty$ almost surely. Fix $s_0\in S$, and define $Y:\Omega \to S$ by letting $Y(\omega)$ be the unique common value of the constant map $\Phi_{-\tau(\omega),0}(\omega)$ when $\tau(\omega)<\infty$, and by setting $Y(\omega)=s_0$ when $\tau(\omega)=\infty$. Then, for every $y\in S$, $\mathbb{P}(Y=y)=\pi(y)$.