Ente Auth stores TOTP codes with end-to-end encryption and syncs them across Android, iPhone, desktop, and the web. It comes from the same people as Ente Photos. You can keep codes on one device without an account. Sync means an account, and the codes are still encrypted with keys you hold. They have published audits.
A second device is the reason to sync. It is also a second place a thief can open if your Ente password is weak. Turn on the app lock. Export an offline backup anyway. SMS codes are not this product. If a site only offers SMS, Ente cannot invent a TOTP secret the site does not support.
Use it when more than one phone needs the same codes. ente.io/auth
Ente Auth’s offline export is the copy that survives Ente. Make one. The sync account should not use a password you also use on the sites the codes protect. If you add SMS-only banks, they simply will not appear. That is a useful inventory of accounts that are still one stolen phone number away from takeover. Fix those at the bank, not in the authenticator.