To make something characteristic of Birmingham; to transform something in the style or manner typical of Birmingham (a city in England or Alabama).
From the place name 'Birmingham' plus the productive verb-forming suffix '-ize,' which creates verbs from nouns. The suffix comes from Greek '-izein.' Place name verbs are rare but do occur.
Creating verbs from city names is extremely unusual in English—we don't usually 'new-york-ize' things! This shows how English speakers will creatively use the '-ize' suffix to coin new verbs when they need them, even if they don't stick around.
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