I have a vague idea. Would Watchtower, Spy, Alchemist, and Scheme count, for example?
None of those are correct.
Here comes the hint:
City
Hunting Party
Beggar
Black Market
are on the list
And you guys can choose your next hint:
1. All the missing cards in the base set and Intrigue
2. A hint on the criteria itself
3. One of the honorable mentions
EDIT: btw, feel free to post thoughts even if you don't have a solid idea