As a native speaker of English, I had never heard the "fully" version until recently. Now I seem to hear it a lot, but only from non-native speakers. Are the two equally acceptable in semi-formal writing?
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Also as a native speaker, I've never heard "full fledged", only "fully fledged" so I suppose we're even.
"Full" as an adverb feels a bit archaic, only generally used in stock phrases such as "full in the face", "know full well" -- although of course "full fledged" could be one such phrase.
NGram says that "full fledged" became more popular than "fully fledged" between ~1910 and 1940, but that before then and overwhelmingly since, "fully" is more common.
Additionally, dictionary.com says that the adverb "full" means "exactly or directly" and "fully" means "completely or entirely". By those definitions, "fully fledged" is the only phrase of the two that really makes sense.
I feel the picture gets more clear if you see this graph: This is for American English only.
whereas, for British English:
I am more comfortable with full-fledged, as that is what I have heard more. Also, there is a large number of people who sill use it quite frequently. Also, as the graph shows in more than one time of the past both have been used with almost same popularity in American English, e.g. 1960, 1900, 1870 (approx.), 1880, 1845 (approx.) etc., though as Slim has shown, there was a gap overall, the major contribution being from the British English towards fully-fledged.
Though, this doesn't make things even, but at least shows the difference in bias, in two variants of the language.