It's rather unfortunate that tab is also used for keyboard navigation between fields, since this would be the default behavior of text areas I would like.
Indeed. Many of the commenters have suggested alternatives, such as using Ctrl-[ and Ctrl-] instead, but the best by far that I've seen would be also binding a listener to the "escape" key, so that tab functionality is restored.
I believe I'll probably end up doing that, unless somebody else does it before I do.