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As I understand Bulgarian and Romania citizens who are married or in civil partnership with UK citizens automatically get unrestricted access to UK work market. But what if they want to get divorced? Does that cancel their status? As you know, UK doesn't require people who have unrestricted access to register. So what happens then? Can you just not notify anybody, and continue working and living here?

In the circumstances you mentioned above, the person will cease from qualifying Under Regulation 2(6) of The Accession (Immigration Worker Authorisation) Regulation 2006, Except if Regulation 2(4) applies.