Multi-height distribution of rational points of split toric stacks

We study the distribution of rational points of split toric stacks with all heights bounded over $\mathbf{Q}$ by lifting the counting problem to an extended universal torsor under the torus associated with the orbifold Picard group. To achieve this, we prove the existence of an integral parametrization of rational points on toric stacks, which allows us to define a lift of the stacky height to this extended universal torsor. This allows us to define the Tamagawa number of a toric stack $X$ as an Euler product and, for a prime number $p$, to interpret the $p$-adic factor via a mass formula counting $\mathbf{F}_p$-points of the sectors of $X$.