Hello @ekemaed,
Because you mention that the marriage registration office wants to verify your documents, I assume that you and your wife did not get married in Germany or one other EU country. If the marriage done abroad is effective and recognized in Germany, you can apply for a residence permit for family reunion as a husband of a German citizen at the immigration office. Information on the required documents can be found under the following link of the immigration office Berlin
****://***.berlin.de/labo/willkommen-in-berlin/dienstleistungen/service.245714.php/dienstleistung/305289/en/
If not, you have to apply for extending your student residence Permit then.
Best regards
@WKZB1