Given a (projective) hyperplane arrangement $\mathcal{A}$ in $\mathbb{C}^n$, in which we assume the intersection of all hyperplanes is $0$, and a building set $G$, De Concini and Procesi define the wonderful compactification $Y_{\mathcal{A},G}$ of $\mathcal{A}$ as the closure of its image in $$\mathbb{P}(\mathbb{C}^n)\times\prod_{X\in G}\mathbb{P}(\mathbb{C}^n/X).$$
It is known, see for example Section 5, that there is a minimal projective toric variety $X_{\mathcal{A},G}$ containing $Y_{\mathcal{A},G}$. In some cases, say for $\mathcal{A}$ the braid arrangement and $G$ the minimal building set, it is known that the wonderful compactification itself is toric.
Is there a general condition on $\mathcal{A}$ and $G$ capturing when the associated wonderful compactification is a toric variety?
