Well, I think other people find it an odd issue, as I do, so that is some comfort. There seem to be two quite clear concepts, and they don’t really contradict each other, in my view.
The first concept is that the rules are very clear that it is the presence of an ally in the right place that provides a flanking bonus, and the perception of the foe is irrelevant. This seems to me to essentially be true.
The second concept is that the rule is terribly counter-intuitive, and should be adjusted slightly so that the foe’s perception of two allies (as with an illusion) grants a flanking bonus, while the mere presence of two allies does not (hence the invisible non-flanker). I think this position is also a pretty sound one...
The ROTG article, part three on sneak attacks, clearly took position two, in the process seeming to imply that it was quoting the rules, rather than creating a slightly different one. The suggested change, by the way, was:
“You get a flanking bonus from any ally your foe can see (and who is in the correct position to flank). If your foe can't see you, you don't provide a flanking bonus to any ally. You literally cannot flank a blind creature; however, a blind creature loses its Dexterity bonus to Armor Class against your attacks (so you can sneak attack it), and you get a +2 to attack it to boot. Creatures with the blindsight ability effectively "see" within blindsight range and can be flanked.”
This is referred to as though it were already part of the rules (which I don’t think is supportable), unlike the section in part four that was clearly labeled as a “Totally Unofficial Rule for Dealing with Foes Trying to Flank You”.
So, I think that by the rules, no flanking illusions, and flanking invisibles. With a little logic massaging, probably the opposite, but probably not by the rules (despite the ROTG endorsement). It kind of seems like something the game might have included a sentence on, to me. Thanks for the posts, they are all really interesting!