Please feel free to edit the question and write it in a more understandable way

Very nice example - thanks Wlodzimierz!

Because it ends with not minimal, and the title says "minimality condition"? :)

That's correct bof: the family of all independent sets of some graph $G=(V,E)$ does form a flag complex. On the other hand, given a flag complex, I would have to think about the question whether there is a graph whose collection of independent sets gives back the edge set of the original flag complex.

Thanks Wlodimierz - I deleted my comment and have recorded your information, you can delete yours now too

Just a short answer: the question is an end to itself; I am toying around with (strongly) minimal coverings in hypergraphs in general.

Good point. For finite $V$ there is always a strongly minimal cover in the setting above. So if there is an example without strongly minimal cover, $V$ must be infinite.