As far as I know S3 itself doesn't support HTTPS (unless you are using CloudFront as a CDN in front of it), might it be the problem in your case? Which url are you trying to access exactly?
So we cracked it. My colleague changed some settings at Amazon, I don't know what, and I added crossorigin="anonymous" to every <img>. I'll be completely honest and say I'd never heard of that attribute before this.