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Let $\mathcal{C}$ denote the family of all coherent distributions on the unitsquare $[0,1]^2$, i.e. all those probability measures $\mu$ for which thereexists a random vector $(X,Y)\sim \mu$, a pair $(\mathcal{G},\mathcal{H})$ of$\sigma$-fields and an event $E$ such that $X=\mathbb{P}(E|\mathcal{G})$,$Y=\mathbb{P}(E|\mathcal{H})$ almost surely. In this paper we examine the set$\mathrm{ext}(\mathcal{C})$ of extreme points of $\mathcal{C}$ and provide itsgeneral characterisation. Moreover, we establish several structural propertiesof finitely-supported elements of $\mathrm{ext}(\mathcal{C})$. We apply theseresults to obtain the asymptotic sharp bound $$\lim_{\alpha \to \infty}\alpha\cdot \Big(\sup_{(X,Y)\in \mathcal{C}}\mathbb{E}|X-Y|^{\alpha}\Big) =\frac{2}{e}.$$

Coherent distributions on the square $\unicode{x2013}$ extreme points and asymptotics