When the Moshiach establishes the messianic kingdom of Yisroel, what type of halacha will come into force over the nation? Sephardi, Ashkenazi, Mizrachi, psak Rambam, or something else? Or will multiple shittot apply to different groups (tribes?) of Yehudim, as is the case in today's Medinat Yisroel?
It's a very good question. If the Igeret of Sherira Gaon is considered, it states clearly that there was no disagreement in regard to halacha originally. That only happened later and as a consequence of a lack of learning.
Part of the redemption and Moshiach's job is to "repair the breaches" within the Torah. Mishnah Torah, Hilchot Melachim 11:1, 11:4
But that said, each individual is required to follow the teachings of his father and grandfather. This is related to keeping minhagim. See: http://www.yeshiva.co/ask/?id=7215
The Lubavitcher Rebbe explains that part of what Moshiach will do is to reveal a higher level of Torah, meaning a deeper understanding in which the varying practices will be seen to be one Torah. This is the idea of "Torah me'iti tetzei." (Torah will come forth from me.) Isaiah 51:4.
And this is not really such a surprise. An example of this already exists in regard to the inauguration of the Mikdash by Moshe Rabbeinu. Each of the Princes of the tribes brought the offering for their tribe on their assigned day and brought exactly the same offering. But the intention of each tribe was different. Bamidbar Rabbah 13:14