The Military Papers is an item found in Omsk, which can be used to travel through the port of Vladivostok without being detained by the authorities. The papers can also be used to speed up travel from Vladivostok or Pyongyang to Yokohama via the Korean Ship.
To obtain this item, Fogg and Passepartout should stay the night in Omsk and then go and visit the governor's residence. After a chat with him for a few hours about Omsk independence, he will ask where the pair are headed. If Passepartout tells him that they are headed east he will point out that Vladivostok, the last stop on the Trans-Siberian, is a military port currently closed to the public. Asking him for help will result in him signing these papers for the gentlemen to guarantee them safe passage through Vladivostok before bidding them farewell.
To speed up travel, you have to explore Vladivostok and scold someone for searching you. However, if Passepartout submits quietly to the search, the guard will inform him for future reference that showing the papers will be useful to speed up said travel (e.g. on said Korean Ship).