This is a modification of Sequential Proportional Approval Voting (either the one by Thiele or Phragmén can be used as the starting point for this).
The ballot allows to put candidates in the categories whitelist, greylist, blacklist. The approvals for the candidates in the whitelist are counted as usual. The other stuff is explained on the ballot "Activate approvals on greylist after __ candidates on the blacklist received a seat."
Like (S)PAV this satisfies the Independence of Irrelevant Alternatives Criterion. But don't get too excited. Imagine you belong to a minority who has a favourite that is on most others blacklists. Electing him will probably activate many greylist approvals. Imagine also that those candidates widely regarded as not great but ok (=greylist material for most) are people your small group strongly hates. Your action might get your favourite elected, but also one person you hate or even several of those.
So one can picture situations when filling out the ballot honestly is worse for some voters than not voting at all. Isn't that odd, satisfying IIA while failing the Participation Criterion?