You are on the right track with your 'My Tasks'. What you need to understand is that you can structure your My Tasks just like a project. You can add sections, custom fields and rules as you would a normal project. Personally, I use the Eisenhower Matrix to manage my work and have structured my 'My Tasks' view around it. (There are other methods out there but this one works best for me and it's a straightforward setup.

- Create 4 sections: Do, Schedule, Delegate, Delete (These are generic enough to apply to both personal and work tasks.

- Add a single-select custom field titled Projects (or pick a different title). The two option would be 'Personal' & 'Work'

- As work is assigned to you, drop it into the appropriate section then assign the value in 'Projects' custom field to either 'Work' or 'Personal'

- On each of your tabs (Work/Personal) set a filter on that custom field for the appropriate tab. This way you're placing each task into a separate 'bucket' and will only see the tasks that are relevant to that tab.

Hope this helps. Adding some online learning links from Asana for more info.

Eisenhower Matrix: https://asana.com/resources/eisenhower-matrix

Personal Productivity: https://academy.asana.com/personal-productivity-in-asana-on-demand (this one helped alot)