Welcome, @vdles,
Multi-homing a task puts a task in multiple projects, but it’s the same task, so it just has a single completion status.
My typical solution when I have a process with multiple steps carried out by different teams/people, and I want them to be able to be grouped together, is to treat Design Work and Social Work as two subtasks of a parent task representing the project as a whole.
But it’s hard to say what the right solution is for your situation without knowing more and spending more time (which is what I do in consulting engagements).
You might find this workshop I did helpful generally in approaching workflows:
Hope that helps,
Larry