We analyse the globular cluster (GC) systems of a sample of 15 massive, compact early-type galaxies (ETGs), 13 of which have already been identified as good relic galaxy candidates on the basis of their compact morphologies, old stellar populations and stellar kinematics. These relic galaxy candidates are likely the nearby counterparts of high-redshift red nugget galaxies. Using F814W (≍I) and F160W (≍H) data from the WFC3 camara onboard the Hubble Space Telescope, we determine the total number, luminosity function, specific frequency, colour, and spatial distribution of the GC systems (GCSs). We find lower specific frequencies (SN < 2.5 with a median of SN = 1) than ETGs of comparable mass. This is consistent with a scenario of rapid, early dissipative formation, with relatively low levels of accretion of low-mass, high-SN satellites. The GC half-number radii are compact, but follow the relations found in normal ETGs. We identify an anticorrelation between the specific angular momentum (λR) of the host galaxy and the (I - H) colour distribution width of their GCSs. Assuming that λR provides a measure of the degree of dissipation in massive ETGs, we suggest that the (I - H) colour distribution width can be used as a proxy for the degree of complexity of the accretion histories in these systems.