  Temperature measurements of galaxy clusters are used to determine their
masses, which in turn are used to determine cosmological parameters. However,
systematic differences between the temperatures measured by different
telescopes imply a significant source of systematic uncertainty on such mass
estimates. We perform the first systematic comparison between cluster
temperatures measured with Chandra and NuSTAR. This provides a useful
contribution to the effort of cross-calibrating cluster temperatures due to the
harder response of NuSTAR compared with most other observatories. We measure
average temperatures for 8 clusters observed with NuSTAR and Chandra. We fit
the NuSTAR spectra in a hard (3-10 keV) energy band, and the Chandra spectra in
both the hard and a broad (0.6-9 keV) band. We fit a power-law
cross-calibration model to the resulting temperatures. At a Chandra temperature
of 10 keV, the average NuSTAR temperature was $(10.5 \pm 3.7)\%$ and $(15.7 \pm
4.6)\%$ lower than Chandra for the broad and hard band fits respectively. We
explored the impact of systematics from background modelling and multiphase
temperature structure of the clusters, and found that these did not affect our
results. Our sample are primarily merging clusters with complex thermal
structures so are not ideal calibration targets. However, given the harder
response of NuSTAR it would be expected to measure a higher average temperature
than Chandra for a non-isothermal cluster, so we interpret our measurement as a
lower limit on the difference in temperatures between NuSTAR and Chandra.
