Each country has its own rules. You should consult yours to find out more. It can be done through the embassy, when possible, or if not via relatives and friends back there. Some countries' consulates have very hostile attitudes towards their own citizens. Their personnel is rule and unresponsive. If this is the case then use the second option I outlined. For US purposes, it makes no difference whether you do that or not.
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Yes, you should change your name on your passport and all other documents. If you don't want to have problems with reentry your identification documents should reflect your legal name.
It is always best to have all documents match.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.