How can I sync an animated gif from Photos.app on a Mac to an iPhone? I have some animated gifs in the Photos library on my Mac. When I sync them to my iPhone (using iTunes) they either don't transfer at all, or they show up as jpegs of the first frame of the gif.

I've tried sending them to myself using Messages.app. This works, but they get shrunk to a postage stamp size that looks small even on the phone screen. I want a way to send them full size, and to keep them synced with my Photos library.
Apple Photos appears to support GIF but not animated GIF.

https://support.apple.com/en-us/HT204264

I suggest you try a different cloud solution for your animated GIFs like Dropbox or Google Drive.