We analysed red giant branch stars in 16 Galactic globular clusters, computing their atmospheric parameters both from the photometry and from excitation and ionisation balances. The spectroscopic parameters are lower than the photometric ones and this discrepancy increases with decreasing metallicity, reaching differences of ~350 K in effective temperature and ~1 dex in surface gravity at [Fe/H] ~ -2.5 dex. We demonstrate that the spectroscopic parameters are inconsistent with the position of the stars in the colour-magnitude diagram, providing overly low temperatures and gravities, and predicting that the stars are up to about 2.5 magnitudes brighter than the observed magnitudes. The parameter discrepancy is likely due to inadequacies in the adopted physics; in particular the assumption of a one-dimensional geometry could be the origin of the observed slope between iron abundances and excitation potential that leads to low temperatures. However, the current modelling of 3D/NLTE radiative transfer for giant stars seems to be unable to totally erase this slope. We conclude that the spectroscopic parameters are incorrect for metallicity lower than -1.5 dex and that photometric temperatures and gravities should be adopted for these red giant stars. We provide a simple relation to correct the spectroscopic temperatures in order to put them onto a photometric scale. Based on observations collected at the ESO-VLT under the programs 065.L-0507, 072.D-0507, 073.D-0211, 078.B-0238, 081.B-0900, 083.D-0208, 085.D-0375, 089.D-0094, 093.D-0583, 095.D-0290, 188.D-3002.