According to the FA descriptions of post #439652 and post #428529, the source Fur Affinity posts of which mark the creation of the concept, a beak sheath is a set of form-fitting polymer 'lips' that on contact with saliva causes a temporary chemical bond to the beak while saliva lubricates and 'inflates' the covers to simulate a mammal's lips, protecting their mate's delicate bits from their sharp tomia and producing flavored lubrication.
See also
- topic #31728 - relevant forum thread