I think that you almost have figured out your answer.   
From a usability perspective you have to show **what the user can reach** _(tag names)_ but also **provide feedback** that he/she can select list items from others tag types.   

I think a combination of both would be great on large screens (a tabular table, and a space "selected items").   
But on mobile we don't have so much space so **you could use a _selected items heading_ (the feedback) followed by _tabs_ (the "what can I reach")**.

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What is your opinion?