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The machinery of quantum mechanics is fully capable of describing a singlerealistic world. Here we discuss the converse: in spite of appearances, andindeed numerous claims to the contrary, any quantum mechanical model can bemimicked, up to any finite accuracy, by a completely classical system ofequations. An implication of this observation is that Bell's theorem is notapplicable in the cases considered. This is explained by scrutinising Bell'sassumptions concerning causality, retrocausality, statistical (in-)dependence,and his fear of `conspiracy' (there is no conspiracy in the language used todescribe the deterministic models). The most crucial mechanism for the counterintuitive Bell/CHSH violation is the fact that, regardless the settings chosenby Alice and Bob, the initial state of the system should be a realistic one.The potential importance of our construction in model building is discussed.

Explicit construction of Local Hidden Variables for any quantum theory up to any desired accuracy