Changed surname after marriage, can I travel to Abkhazia with an unchanged passport?

In short, yes, you can. But there are nuances:

For travel to Abkhazia, Russian citizens only need a valid internal Russian passport; a foreign passport is not required. According to the law, the Russian passport remains valid for another 90 days after a surname change, reaching 20 or 45 years of age. So if your trip falls within this period, you can safely go, no problems will arise. If departure or return from Abkhazia is later than 90 days, there may be issues.

For more details on the entry and exit procedures in Abkhazia, what documents are needed, how to cross the border, and everything else, see the detailed guide Visa and entry to Abkhazia

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Follow the link, all these questions are discussed in the chat - https://t.me/abhazia_travel_chat