  Photomultiplier tubes (PMTs) are often used in low-background particle
physics experiments, which rely on an excellent response to single-photon
signals and stable long-term operation. In particular, the Hamamatsu R11410
model is the light sensor of choice for liquid xenon dark matter experiments,
including XENONnT. The same PMT model was also used for the predecessor,
XENON1T, where issues affecting its long-term operation were observed. Here, we
report on an improved PMT testing procedure which ensures optimal performance
in XENONnT. Using both new and upgraded facilities, we tested 368 new PMTs in a
cryogenic xenon environment. We developed new tests targeted at the detection
of light emission and the degradation of the PMT vacuum through small leaks,
which can lead to spurious signals known as afterpulses, both of which were
observed in XENON1T. We exclude the use of 26 of the 368 tested PMTs and
categorise the remainder according to their performance. Given that we have
improved the testing procedure, yet we rejected fewer PMTs, we expect
significantly better PMT performance in XENONnT.
