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There are certainly many situations in which the data you have or are considering can be placed in or represented by different consistent families which are mutually incompatible, and this is one of the standard objections to consistent histories. My response is that there is no “right” family; this is something I take up under the heading of ‘unicity’ in Sec. 27.3 of Ch. 27 of my book [http://quantum.phys.cmu.edu/CQT/]. On the other hand, for particular instances, as when one is concerned with how measurement outcomes reflect the properties the device was built to measure, it turns out that the choice is essentially unique; though making this precise requires something of a technical discussion, as found in Ch. 16 of my book. Or take a look at the discussion of the ‘second measurement problem’ in arXiv:1501.04813.
I hope these comments are of some help.
Best wishes! Bob Griffiths