I don't know the answer
If someone allows a reporter to use his quote only if he isn't named, the reporter will often write that he is speaking on condition of anonymity.
This sounds wrong to me, and I always want to add a "the" -- he spoke on the condition of anonymity.
But I can find several other examples that use this phrasing: on condition of reciprocity, on condition of repentance, on condition of good behavior, on condition of progress. So it's not just a journalism cliche.
Is this a phrase I'm not familiar with? Does anyone have an answer better than "it sounds good"?
UPDATE: I should mention that Bill Walsh covers this in "Lapsing Into a Comma," preferring, as I do, the the. He says on condition of anonymity is "grammatically flawed." But he doesn't explain how.