When my partner and I moved in together, sharing responsibility for chores didn't come naturally. It's not that we're irresponsible, we clean up after ourselves and if something has to be done, we'd do it. The problem was the chores you do so they don't mount up later, things like laundry, cleaning the bathroom, changing bed sheets, etc.
I started by searching for an app that could help us maintain a routine.
The solution would allow us to make tasks repeat with very specific criteria (eg. dust surfaces on the last Sunday of every month unless they've been dusted recently). It would also need to be collaborative – when I add something, my partner sees it, and when my partner marks something as complete, I see it's done.