We're building a new multiplayer editor for tasks/notes [1] which supports both text and outliner operations. Although it behaves like a flat text document, the outliner features essentially turn the document into a large tree under the hood. We do something similar to the highly-available move operation to sync changes:

There is one operation to change the tree, called insmov (move-or-insert). Whenever a client is online it can sync changes C to a server. Whenever the server has remote changes for us, it will send us back a list R of all changes since our last sync in a global linear order. We then undo any of the insmovs in our changeset C, and (re)apply all changes in R + any new changes we didn't sync yet.

We don't use any fractional indices though. Instead, our insmov tuple not only contains a parent P, but also a previous sibling guid A. Because all tree ops will eventually be applied in the global linear order as determined by the server, "sorting" is handled by just using the insmov operation.

Most of the time the undo'ing of operations is not needed though. Only when the server has insmov changes we don't know about while we are sending new insmovs ourselves do we need to ensure we replay the operations in the correct order. That's likely to happen when you reconnect to wifi after a long flight, but not so likely when updates are pushed in real-time over websocket when you're online (plus it's not needed for non-insmov operations, like updating text).

[1] https://thymer.com