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Visas are a very complex and ever changing subject and you are strongly advised to contact RCI on that matter - also it all depends on your nationality and what passport you have. If you are lucky enough to have a passport for more than one country - i.e. resident in the EU and maybe born in Australasia for instance, I would advise that you take all you have with you - assuming of course that they are all in the same name!
Having said all that, my wife and I, who hold British Citizen passports, travelled to the Far East on the QE2 in 2003, which was part of the world cruise she did and visited Papau New Guinea, Guam, Japan, Taiwan and Hong Kong. The following year we were on the Pacific Princess calling at Thailand, Vietnam, Singapore and Malaysia before cruising on to India and Africa. For Vietnam the cruise line sold us tourist visas whilst we were aboard and charged our on-board account. All other necessary visas were arranged by both cruise lines.
It should also be noted that having a visa is no guarantee that the country will allow you to go ashore. We were on a Costa Fortuna cruise last Christmas and called into Libya. Of course all of the 3,000 passengers were excited as General Gadafi had only recently allowed tourists back to his country. My wife and I spend a fantastic day visiting Roman ruins that are some of the finest in the world. When we returned to the ship we found out that the Libyan authorities had refused entry to all Americans, Canadians and Australians aboard - even though they had all got obtained special visas from their local Libyan embassies. In other countries the visa requirements are as flexible as a piece of plastic and depend on who is on duty that day. Egypts official stance on visas for instance is that UK residents need to apply in person to the consulate on one day and pick up the visa the next. In practice you can obtain a visa at the point of entry as long as you have foreign cash - a few years ago we flew into Luxor Airport and each passenger was told to have a £10 handy for the visa. At arrival it was bedlum and an official merely took the £10 note and waived us through, not even looking at our passports.
So you see, it is always wise to contact your carrier - in your case RCI, and have something in writing from them about visa needs. They may refuse to commit themselves and tell you to contact each country yourself. I am sure everything will be fine and you have a great cruise, just remember, if crossing the equater and going from the Southern Hemisphere to the Northern Hemisphere or vica versa, the weather can be very hot in one and very cold in the other. On the QE2 it was very hot and sticky in Sydney (visa needed!)in February (record 46c) and remained hot until Tokyo, Japan where it was snowing with a sub zero temperature. By the time we entered Hong Kong in early March the temperature had risen to 30c.