morphophoneme

One example in English where a morphophoneme could be postulated is in plurals that are accompanied by sound change, such as the common alternation between /f/ singular and /vz/ plural: 'knife' /naif/, plural 'knives' /naivz/.

If this was completely regular you could have a simple rule that the phoneme /f/ changed to /v/ before the plural morpheme /z/. The grammar level influences the sound level but is not mixed up with it.

However, the sound change is not entirely regular. A few words like 'fife' exist that do not take part in the alternation: plural 'fifes', not *'fives'. Moreover, you cannot make a reverse rule either, one which says an underlying /v/ is pronounced /f/ when not followed by plural, because there are cases where that doesn't apply either: the singular of 'hives' is not *'hife' but 'hive'.

At this point you can invoke a morphophoneme, usually symbolized by some compromise symbol such as /F/, and say it's a unit that behaves phonetically like an /f/ in the singular but like a /v/ in the plural. This means that you have three different underlying patterns, all regular:

    /faif/ + PL -> /faifs/
    /haiv/ + PL -> /haivz/
    /naiF/ + PL -> /naiFz/

I think morphophonemes are a silly idea myself.