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Our current understanding of the Universe depends on the interplay of several distinct matter components, which interact mainly through gravity, and electromagnetic radiation. The nature of the different components, and possible interactions, tends to be based on the notion of coupled perfect fluids (or scalar fields). This approach is somewhat naive, especially if one wants to be able to consider issues involving heat flow, dissipative mechanisms, or Bose-Einstein condensation of dark matter. We argue that a more natural starting point would be the multipurpose variational relativistic multifluid system that has so far mainly been applied to neutron star astrophysics. As an illustration of the fundamental principles involved, we develop the formalism for determining the nonlinear cosmological solutions to the Einstein equations for a general relativistic two-fluid model for a coupled system of matter (nonzero rest mass) and radiation (zero rest mass). The two fluids are allowed to interpenetrate and exhibit a relative flow with respect to each other, implying, in general, an anisotropic Universe. We use initial conditions such that the massless fluid flux dominates early on so that the situation is effectively that of a single-fluid and one has the usual Friedmann-Lemaître-Robertson-Walker spacetime. We find that there is a Bianchi I transition epoch out of which the matter flux dominates. The situation is then effectively that of a single fluid and the spacetime evolves towards the Friedmann-Lemaître-Robertson-Walker form. Such a transition opens up the possibility of imprinting observable consequences at the specific scale corresponding to the transition time. [ABSTRACT FROM AUTHOR]