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First it is not a "false name" if it is merely a mis spelling of your real name,    The issue may be why you did not correct your passport which is suppose to be the correct government issues ID which requires a documents to get it. 
So if there was an error. 
 
If you were to actually use a false name, but you used your name as registered with the federal government 
 
You in this case are legally married,  and will have to get a divorce.  You will need to correct your passport. 
 
A big difference in false name, to a mis-spelled name
     
     
    
    
    
    
    
    
  
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