  The three tachyon-free non-supersymmetric string theories in ten dimensions
provide a handle on quantum gravity away from the supersymmetric lamppost.
However, they have not been shown to be fully consistent; although local
anomalies cancel due to versions of the Green-Schwarz mechanism, there could be
global anomalies, not cancelled by the Green-Schwarz mechanism, that could
become fatal pathologies. We compute the twisted string bordism groups that
control these anomalies via the Adams spectral sequence, showing that they
vanish completely in two out of three cases (Sugimoto and $SO(16)^2$) and
showing a partial vanishing also in the third (Sagnotti 0'B model). We also
compute lower-dimensional bordism groups of the non-supersymmetric string
theories, which are of interest to the classification of branes in these
theories via the Cobordism Conjecture. We propose a worldvolume content based
on anomaly inflow for the $SO(16)^2$ NS5-brane, and discuss subtleties related
to the torsion part of the Bianchi identity. As a byproduct of our techniques
and analysis, we also reprove that the outer $\mathbb{Z}_2$ automorphism
swapping the two $E_8$ factors in the supersymmetric heterotic string is also
non-anomalous.
