Bernoulli coupling by a shared uniform random variable ($\mathbb{P}(B_r\ne B_s)=|r-s|$)

For $r,s\in [0,1]$ and a random variable $U$ with the uniform distribution on $[0,1]$, set $B_r=1_{\{U\le r\}}$ and $B_s=1_{\{U\le s\}}$. Each has the corresponding Bernoulli distribution, and $\mathbb{P}(B_r\ne B_s)=|r-s|$, because disagreement occurs precisely between the two thresholds. This is a maximal coupling of the two laws.