For example,
I VOTE:
[_] LEAVE [_] REMAIN:
My choice above means (choose one): (_) more welfare; (_) less welfare. Foreign trade will be (choose one): (_) simplified; (_) complicated. etc.
If the quiz choices don't match the vote itself, the vote is INVALID.
Thus the amount of uninformed votes would be reduced to zero.
Of course the options would have to be agreed upon by both sides.
"Oh, but someone could easily be taught to 'cheat', you could tell someone what answers to pick!" Well, sure. Good luck convincing an immigrant to tick the "(_) less jobs for immigrants" box.