Let $(X,B)$ be a klt pair where $K_X+B$ is $\mathbb{R}$-Cartier. Let $\pi:X\rightarrow U$ be a projective morphism of quasi-projective varieties. Assume either that $B$ is $\pi$-big and $K_X+B$ is $\pi$-pseudo-effective, or $K_X+B$ is $\pi$-big, then the work of Birkar-Cascini-Hacon-McKernan proves that the log canonical ring $$\mathfrak{R}(\pi,K_X+B)=\bigoplus_{m\in\mathbb{N}}\pi_*\mathcal{O}_X(\left\lfloor m(K_X+B)\right\rfloor )$$ is a finitely generated $\mathcal{O}_U$-algebra under the assumption that $K_X+B$ is $\mathbb{Q}$-Cartier.
I wonder why can't we derive the same conclusion when $K_X+B$ is just $\mathbb{R}$-Cartier? After all, the coefficients of $B$ should cause no trouble after being rounded down, and $\mathfrak{R}(\pi,K_X+B)$ should be the same as when $K_X+B$ is $\mathbb{Q}$-Cartier. What am I missing?