iPhone shows 12,000 photos but iCloud Photos only shows 9,000, even though backup is complete

When using iCloud Photos, none of those are in the iCloud backup, so having the backup completed does not mean your photos have finished syncing to iCloud. You should be able to view the syncing status of the photos by going to the Photos app with the information found here:

On your iPhone or iPad: Open the Photos app, then tap the profile button in the upper corner (in iOS 26 or later, go to Collections, then tap the profile button). Check your library status below your name and photo count. In iOS 17 or earlier, go to the Library tab, tap All Photos, and scroll to the bottom to see your library status.

If your iCloud Photos are not syncing - Apple Support

wajeehaslam wrote: … My iPhone shows around 12,000 photos, while iCloud Photos only shows about 9,000.

Where do you see these numbers? Pictures can be in places other than in Photos, so it's hard to know what you mean. Do you use iCloud Photos?

My iPhone is fully backed up to iCloud,

If you mean you use "iCloud Backup," then as Mac Jim ID says, that doesn't include Photos (if you use iCloud Photos.) If you mean iCloud Photos, then that is not a backup. Rather, iCloud Photos is a sync, or synchronization. Full synced would be indicated at the bottom of Photos on a Mac like this:

or on a phone like this:

It's hard to know what you mean unless you tell us what you actually see rather than what you infer in means. Let us know…

Some pictures may be transferred to a phone outside of Photos' control, through some flaky app, for instance. Those pictures can't be edited or deleted or copied to Photos. Do you have pictures like that-- pictures that can't be edited? Usually those are all erased when first connecting to Photos.