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Consistency relates to showing, not breeding. My understanding, based on the wording of PT score results, is that consistency is basically the horse's moodiness. An even-tempered horse is more likely to have fairly consistent show results, whereas a moody or temperamental horse is more likely to have inconsistent show results. Very inconsistent seems to mean very bad tempered.
While consistency is fairly random, inconsistent horses do tend to produce more inconsistent and very inconsistent foals, much like a bad tempered stallion will likely produce more ill-tempered or irritable foals than other studs tend to produce.

Edit: consistency is based on hidden numbers, so there is a range involved. When a horse falls too far into the inconsistent end of that range, breeding advice or strict breeding advice will spay/geld them. I believe 'Very Inconsistent' horses will always be spelded by this testing. They're usually the ones that get the "this horse has a flighty temperament" message from BA or SBA.

Thank you both - Ammit, so even if both parents are consistent, there is still an average hidden number meaning they can be lower on the consistent scale and produce inconsistent foals?