  We obtain new constraints on the anomaly coefficients of 6D
$\mathcal{N}=(1,0)$ supergravity theories using local and global anomaly
cancellation conditions. We show how these constraints can be strengthened if
we assume that the theory is well-defined on any spin space-time with an
arbitrary gauge bundle. We distinguish the constraints depending on the gauge
algebra only from those depending on the global structure of the gauge group.
Our main constraint states that the coefficients of the anomaly polynomial for
the gauge group $G$ should be an element of $2 H^4(BG;\mathbb{Z}) \otimes
\Lambda_S$ where $\Lambda_S$ is the unimodular string charge lattice. We show
that the constraints in their strongest form are realized in F-theory
compactifications. In the process, we identify the cocharacter lattice, which
determines the global structure of the gauge group, within the homology lattice
of the compactification manifold.
