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You can use 3rd party password managers to sync your passkeys however you please. Apple has no access to keychain data: https://support.apple.com/en-us/HT213305
Could you give an example? That support article doesn't mention anything about it. I do recall reading something about that recently, but I read the article and it it made no mention of what syncing passkeys actually means. It sounds like theoretically I could write an app that uses the same APIs Apple uses to sync encrypted credentials between TPMs. But "however you please" sounds suspiciously like "however you plea…
You login to iCloud which gives your device write access to iCloud storage. The device creates a private key inside the TPM, and uploads the public key. When you add a new device, one of your existing devices has to use its hardware key to sign your new devices key. It also must be signed by a second key derived from your iCloud password (so you). Apple doesn't go into the details on this, but I believe the originating TPM also signs the key with a chain of trust that goes back to Apple itself.
The keychain is encrypted locally and exported to iCloud. The key used to encrypt the keychain is also encrypted using the public keys of all trusted device keys in iCloud that have both a signature from another trusted device and from you.
As a user you can also just open up Keychain Access on a Mac and ask it to generate a CSV export of your entire iCloud keychain - which is how you would go about migrating to another password manager.