It turns out we had another 16.04 machine which was working fine. But as
soon as I updated its sudo from 1.8.16-0ubuntu1.2 to 1.8.16-0ubuntu1.3,
it stopped working too.
So it looks like I have a reproducing case for this and I can
investigate further - I suspect it's a behaviour change from this fix:
https://bugs.launchpad.net/ubuntu/+source/sudo/+bug/1607666
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