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This sounds like Google Docs, perhaps combined with Trello (for the Kanban board and calendar view), would do for you. It's got live updating -- I don't think it tracks edits by author exactly, so...
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#3: Attribution notice added
Source: https://writers.stackexchange.com/a/45347 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#2: Initial revision
This sounds like Google Docs, perhaps combined with Trello (for the Kanban board and calendar view), would do for you. It's got live updating -- I don't think it tracks edits by author exactly, so you could agree on a convention -- like my students on one team devised a rule where each of them owned 2 colors. (4 students, so like light blue and dark blue was one, red and pink was another, etc.) Anything NEW they would do in one of their colors. Any EDITS to others' work, they'd do in their edit-color. (They copied original sentence to comments, in case people wanted to compare.) They would comment each paragraph with "ok" if they had no changes. Their teamleader would turn paragraphs BLACK if everyone had said "ok". Since it's only 2 of you, that part could be simpler. If you want to stay in Google things, you could fake a KanBan with Google Sheets, just name your columns, and each spreadsheet Cell becomes an Item, which you can move through the process. And then Google Calendar of course also exists for deadlines or blocking out time to write. But Trello is free to use and lets you link to Gdocs also, so that's probably a better Kanban/Deadline solution.