We use data from the HST Coma Cluster Treasury program to assess the richness of the Globular Cluster Systems (GCSs) of 54 Coma ultra-diffuse galaxies (UDGs), and hence to constrain the virial masses of their haloes. For 18 of these the half-light radius exceeds 1.5 kpc. We use a maximum-likelihood method to take account of the high contamination levels. UDG GCSs are poor: for 14 of the largest 18, $N_{GC}