Let $X=(X_1,X_2,X_3)\sim \text{Dirichlet}(a_1,a_2,a_3)$ and $Y=(Y_1,Y_2,Y_3)\sim \text{Dirichlet}(a_1+b_1,a_2+b_2,a_3)$, where all $a_i$ and $b_i$ are positive. Is there a natural coupling between $X$ and $Y$ such that $X_1\geq Y_1$ and $X_2\geq Y_2$ with probability 1?

The following coupling does not guarantee this property: Define independent random variables $G_{c}\sim \text{Gamma}(c)$ for $c\in\{a_1,a_2,a_3,b_1,b_2\}$, and let

$$X_i = \frac{G_{a_i}}{G_{a_1}+G_{a_2}+G_{a_3}} $$

and

$$Y_i = \frac{G_{a_i}+G_{b_i}\mathbb{I}(i\in\{1,2\})}{G_{a_1}+G_{b_1}+G_{a_2}+G_{b_2}+G_{a_3}}. $$