According to Pearce (1992), Fudenberg, Levine and Maskin (1994) "The Folk Theorem in Repeated Games with Imperfect Public Information" Econometrica, 62 represent the state of art in discounted folk theorems for a broad range of information structure, and "anyone interested in repeated games should read it closely".

It is difficult to state their folk theorems without relying on mathematical symbols. Here, I try to summarize the key insight as I quote the relevant sentences (in italic) from Pearce (1992).
To derive folk theorems, they consider two conditions, the individual full rank condition and the pairwise full rank condition. The former is needed to ensure that a player's different possible actions can be distinguished, and hence encouraged or discouraged. The latter is needed to acquire information which discriminates statistically between deviations by some player and others.
Without the individual full rank condition, it may not be possible to induce players to play some strategy, no matter what rewards are attached to signal realizations. In contrast,
this (="the individual full rank condition") guarantees that any behavior can be enforced if arbitrary continuation payoffs can be used.

The failure of the pairwise full rank condition explains the inefficient results by Radner, Myerson and Maskin (1986).
The problem there was that the only way to enforce good behavior was to punish both players in the event that output is low. Efficient (or nearly efficient) cooperation in a model where no player's actions are observed, generally requires that, when one player's continuation payoff is reduced, another's must be increased; surplus should be passed back and for the amongst players, not thrown away.

With the additional restrictions on the information structure guaranteed by the full rank conditions, FLM prove a folk theorem of virtually the same degree of generality as for perfect monitoring.

As a final remark, it should be noticed that the pairwise full rank condition does not necessarily be satisfied at equilibrium strategy profiles.

It would have been reasonable to guess that, to prove that a desired profile "gamma" can be enforced (almost) efficiently, it would be necessary to impose pairwise full rank relative to deviations form "gamma". By contrast, all that is actually assumed is that, for each "i" and "j", there is some distinguishing "alpha" that allows i's and j's deviations to be distinguished, and not necessarily the same "alpha" for each pair of players! FLM demonstrates that a profile as close as desired to "gamma" can be found that puts a little weight on the strategies used in the "distinguishing profiles," and discriminates as required between deviations of different players.

I should read FLM again...