I think this is a "modo di dire" - naturally I understand the words but not the meaning, please help.
Penso che sia un modo di dire, ma anche se capisco le parole, non sono sicuro del senso, aiuto!!
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"action tu che action io" is a colloquial form to indicate that we both are doing or should do something with a will and at the same time, possibly in (usually friendly) competition: corri tu che corro anch'io = run, and I'm running too.
It is also used for continuous actions: ci siamo visti al pomeriggio, ma, chiacchiera tu che chiacchiero io, si è fatta notte - we met in the afternoon and, one gossip after another, the night fell.
So literally mangia tu che mangio anch'io means we are both eating heartily, or let's go on eating. I have found this exact usage in a parody of a sonnet by Dante by Carlo Lapucci:
Guido i' vorrei che tu, Lapo ed io fossimo presi per incantamento e messi in un tinel, che a piacimento da mangiar desse d'ogne ben di Dio. Ivi sbafando in tre per quattrocento di bere insieme crescesse 'l disio; indi, **mangiate voi che mangio anch'io,** ...
Then there is a figurative sense for mangiare which also means grafting, accepting bribes.
Therefore mangia tu che mangio anch'io could translate to a reciprocal encouragement to accept (or continue accepting) bribes.
The answer might depend on the context, but it probably refers to a few corrupted people (probably politicians or something close), each of which knows that the others are corrupted but is not willing to report them in order to protect their own interests. In the analogy, "to eat" is to gain benefits from corruption: by not preventing others from "eating" (not reporting them) you can keep "eating" as well (because they do not report you).
In the Camilleri's novel mentioned by Charo this is the actual use (the sentence is said about a mayor and other unnamed politicians).