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Google reveals 4.5 million results (archived) of "South African coronavirus variant" and related terms (see below) as of 13 January 2021. Some government/media sources used it as a name, some used it colloquially. These are the names in most common (colloquial) usage, as compared to 501.V2 variant. Below is a shortlist of sources of these terms. denote scientific sources.
The South African variant also has the N501Y mutation and several other mutations but emerged completely independently of the UK strain and is not related to it.