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.
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