It turns out the problems are caused by DPM. Without DPM (using radeon.dpm=0)
everything works as it should.

I have tried tweaking the DPM parameters using
https://wiki.archlinux.org/index.php/ATI#Dynamic_power_management, but changing
dpm_state to `performance` and power_dpm_force_performance_level to `high`
doesn't solve the issues (in fact, changing these parameters seems to have no
effect at all?).

Obviously disabling DPM isn't ideal, so it would be nice if there was a
workaround/fix.