The local-global principle for integral points on stacky curves

We construct a stacky curve of genus $1/2$ (i.e., Euler characteristic $1$) over $mathbb{Z}$ that has an $mathbb{R}$-point and a $mathbb{Z}_p$-point for every prime $p$ but no $mathbb{Z}$-point. This is best possible: we also prove that any stacky curve of genus less than $1/2$ over a ring of $S$-integers of a global field satisfies the local-global principle for integral points.