Details:
A rumor has circulated since around 2016 that Omar’s 2009 marriage to Ahmed Nur Said Elmi (a British-Somali citizen) was actually to her biological brother, purportedly for immigration purposes. �
Business Insider +3
Major fact-checking organisations have reviewed it and found it unverified:
PolitiFact concluded there is “no verified evidence that Elmi is Omar’s sibling.” �
PolitiFact
Snopes has similarly labelled the claim as unproven, noting reliance on anonymous sources and internet forums. �
Business Insider +1
Omar has publicly called the claim “absurd and offensive” and denied it. �
Business Insider +1
Investigations, including by media outlets, did look into the marriage timeline and documentation, but found no legal proof of a sibling relationship or immigration fraud in that context. �
The Times of India +1
Conclusion: The claim that she married her brother is not substantiated by credible evidence. It remains a rumor and appears to be politically motivated.
2) Did she obtain U.S. citizenship on false pretenses (via that alleged marriage)?
Short answer: There is no verified evidence that she gained citizenship through marriage fraud.
Details:
Omar immigrated from Somalia as a refugee (via Kenya) in 1995, and later became a U.S. citizen. �
Newsweek +1
The allegation would imply that the 2009 marriage to Elmi was used to gain immigration benefits for him (or her) via fraud. But:
The key claim — that Elmi is her brother — is unproven.
There is no public record or official finding showing immigration fraud in her case. �
PolitiFact +1
Experts note that while fraudulent naturalization is a basis for denaturalization (“stripping citizenship”), such a case would require a legal finding of fraud. No such case against Omar appears public. �
irli.org +1
Conclusion: There is no verified evidence that Omar obtained U.S. citizenship through a fraudulent marriage or misrepresentation.

My summary for you:
The “married her brother” claim: Unverified, lacks credible proof.
The “got citizenship via false pretense” claim: Unsubstantiated, no legal finding.
The rumours appear to be part of politically charged attacks rather than fact-based conclusions.