The relevant entries from the OED are:
I also had a look at Worcester’s dictionary, primarily to see if he had any citations. I’m not sure about the date, as the first twenty pages are missing, but these are the relevant entries:
So it would seem that auditorial is “established”, while auditorily is not mentioned. Maybe it is a new(ish) word!
Adrian Stenton is a PhD candidate at Leiden University Centre for Linguistics and is currently investigating number concord in the species noun phrase. Adrian is part of the project Bridging the Unbridgeable: linguists, prescriptivists and the general public, which is supervised by Prof Ingrid Tieken-Boon van Ostade.
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Curiouser and curiouser.This does seem to be a changing usage. Could there be some difference between US and UK? On the US side, the Random House Dictionary (1967) lists both auditorily and auditorially as adverbs for auditory. But Merriam-Webster’s Third New International (1971) lists no adverb (and define the adjective auditorial only as related to an audit or auditor). Here is the result from the Google N-gram Viewer, which seems to show a dramatic rise in the use of auditorily between 1950 and 1980, followed by a steep decline since then: https://books.google.com/ngrams/graph?content=auditorily%2C+auditorially&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cauditorily%3B%2Cc0%3B.t1%3B%2Cauditorially%3B%2Cc0