For a good while I didn't realise this was what the problem was; however, when I finally went to Specsavers for an updated prescription, they saw something was wrong and referred me to an ophthalmologist. In due course (about a year; I was lazy about driving the process along, because it was scary) this lead to, well, this (WARNING, unusually graphic images for a Wikipedia surgery article).
Before (note eye orientation):
And here I'm appearing as a rather low-budget Borg with the eye-protection thingy that I must wear when sleeping.
End result; a bit of pain, a fair bit of discomfort (feels like there's something in my eye; this will persist until the sutures in my eyeball dissolve in a week)... and more-or-less working stereoscopic vision! Much though I was dreading it, it really didn't work out badly at all (so far; there's always the possibility that my eyes might just drop out or something, I suppose).