Majority of countries recognise dual nationality and having 2 passports can potentially simplify your travels.
It is fine in most cases to travel to a country on one passport and then enter the next country on a different passport. For instance, one may have a British and an American passport. It is cheaper to apply for a Russian visa with a British passport, but you won't need a visa to travel to Mongolia on an American one. So it is possible to get a Russian visa with a British passport, travel to Russia with it, and then swap it for the US one to enter Mongolia.
The 2 countries you may travel through on the Rally which don't recognise dual nationality are Turkmenistan and Iran. That means you'll need to use the same passport to enter Turkmenistan and Iran as you use to enter countries immediately before and after it (e.g. Turkey and Uzbekistan).
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