A fee, toll, or right to keep and maintain pigeons or doves in a culverhouse; the practice or system of dovekeeping.
From culver (dove) plus -age (a suffix forming nouns of action, practice, or a collection), similar to 'dovecote' + -age. This would be a feudal or medieval right associated with landholding.
In medieval feudalism, the right to own a culverhouse and collect culvertage was a valuable privilege granted by nobles—they controlled pigeons because they controlled an important protein source and a prestige symbol all in one.
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