In classical stochastic theory, the joint probability distributions of a stochasticprocess obey by definition the Kolmogorov consistency conditions.Interpreting such a process as a sequence of physical measurements withprobabilistic outcomes, these conditions reflect that the measurements donot alter the state of the underlying physical system. Prominently, thisassumption has to be abandoned in the context of quantum mechanics, yetthere are also classical processes in which measurements influence themeasured system. Here, conditions that characterize uniquely classicalprocesses that are probed by a reasonable class of such invasivemeasurements are derived. We then analyze under what circumstances suchclassical processes can simulate the statistics arising from quantumprocesses associated with informationally-complete measurements. It isexpected that this investigation will help build a bridge between twofundamental traits of non-classicality, namely, coherence and contextuality.