I haven't traveled the world as in every country or anything, but I have traveled to Europe for extended periods of time and still continued to earn a consistent MLM income. However... this was "after" I had build a large enough team where the income would continue to come in and grow even if I died as long as the team website remained online and Google continued to list it high on the first page of search results. In this case with a 5 level commission program for instance, you could easily have 4 people drop out on the 8th level and 3 join on the 9th and it's likely you'd never know it and your income for that month would remain the same.
So I already know it is possible, but the big question I suspect is whether your friend is saying he can travel the world with a full time MLM income check each month and... what is full time income or is enough income for him? $500 a month, $4000 a month, $7000 a month etc. If he's talking "huge" numbers as in the top 1% of the Distributors in a good company and over $15k a month, then I believe he's in the minority and most of us could not realistically quit our day jobs and travel the world strictly on our MLM income.
I hope that helps some, have a wonderful day there and - great question too!