In most occasions C&P from other websites (on Android you avoid typing)

But check it out, if you switch to the second non alphanumeric symbol set, the BAR button will change to TAB.

And no, pressing doesn't move to next field.

Possible workaround: pre-filtering input with s/\t/ /g

seems like the right thing. I can't imagine anyone expecting TAB to actually do anything else in this context (especially since whatever it might be won't work in regular browsers anyway).