Known experiments with the path entangled photon pairs are considered here under more general conditions widely broadening the domain of used bases. Starting from symmetric beam splitters and equally weighted superposition in the initial setup, we allow arbitrary beam splitters and in addition insert the new elements: absorptive plates. The first innovation allows one to vary the amplitudes of local interferences. The second one enables the experimenter to monitor the nonlocal superposition amplitudes, thus varying the entanglement strength from maximal to zero. The generalized scheme reveals an interesting effect: the local coherence observed for independent photons disappears already at infinitesimally weak entanglement between them. Mathematically, local coherence turns out to be a discontinuous function of entanglement strength. The same features are unveiled for a quite different system, spin entangled fermion pair. We can thus conjecture a general rule of total mutual intolerance between local coherence and entanglement: any local coherence must vanish completely not only at maximal, but even at arbitrarily weak entanglement between members of studied pair. Altogether, the generalized thought experiment shows that coherence transfer is a complicated phenomenon with common features for various bipartite systems and different types of observables. Key words: Bi-photon, bi-fermion, entanglement, correlations, coherence transfer
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