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I am looking for a Chromium-based browser which support sync out of the box.
I want to sync:
With "sync" I mean "sync between desktop (Linux) and Android".
Required Features:
Updated my answer, as you changed the focus from password sync to a more general sync.
I'm going on a limb here, and suggest that an open-source chromium based solution with an out-of-the-box password sync doesn't exist to sync Linux and Android. (One of the reason I abandoned chromium based browsers myself as much as I can) The most complete list of chromium based android projects I know of is probably this one.
As for password sync, I would suggest a different approach: Bitwarden which is not only open-source, it can be completely self-hosted.
Desktop browser integration is done via an extension, and Android integration is on the OS level via the App's auto-fill option.
If you are interested in the self-hosted solution, other than the official solution, there is the slimmer server implementation of vaultwarden.
From my personal experience I can tell you, I've had this installed for a couple of years now, and have my family members (and a few friends) using it as the only password vault. Doesn't require to be a techie to use, integration into Android UI works very well with most browsers and most sites.
As for browser history - that would definitely need some native browser solution - but I'm unclear here - I personally but very little effort into this area.
If you run a nextcloud instance, I think floccus is a great option for bookmark sync, but then again - I don't think it solves you the android client issue.
