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Is git-based syncing something that's built into Obsidian? I'd imagine it's awkward setting up git management on a smartphone outside of Obsidian?

There is a git sync community plugin and it works well on phones.

Is it the plugin « Obsidian Git »?

Yup! Thats the one - works great.

I use Working Copy on my iphone to push / pull from my git repo

Same on laptop

Obsidian vault is just the directory where the git repo is