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Video abstract for the article 'Weakly linked binary mixtures of F = 1 87Rb Bose--Einstein condensates' by M Melé-Messeguer, B Juliá-Díaz, M Guilleumas, A Polls and A Sanpera (M Melé-Messeguer et al 2011 New J. Phys. 13 033012). Read the full article in New Journal of Physics at http://iopscience.iop.org/1367-2630/13/3/033012/fulltext/. GENERAL SCIENTIFIC SUMMARY Introduction and background. Ultracold bosonic clouds of atoms provide a powerful tool for studying complex quantum many-body phenomena with a large degree of control. A quite spectacular new development is the experimental observation of quantum tunneling of large clouds of atoms (~1000 atoms) through potential barriers by the Heidelberg group for a single component Bose--Einstein condensate (BEC). The physics resembles that of superconducting Josephson junctions, with sizeable differences arising from the atom--atom interaction term, which gives rise to so-called self-trapped states. Main results. In the present paper we consider a binary mixture of BECs made by populating the m = ±1 components of an F = 1 spinor condensate. The system is studied by solving the exact three-dimensional (3D) Gross--Pitaevskii equation (GP3D), which is a well-known mean-field equation used for weakly interacting systems at very low temperature. We compare it with the usual models to describe the Josephson dynamics: (I) the standard and improved two-mode equations, which are two different simplifications that describe the system using only the ground and the first excited states, and (II) two different reductions of the dimensionality of the GP3D equation: the 1D Gross--Pitaevskii equation (GP1D) and the non-polynomial non-linear Schrödinger equation (NPSE). Moreover, we have provided a concise and self-contained derivation of the different models used to describe the dynamics, and an exhaustive study of the conditions for the possible regimes of the system. Wider implications. A large variety of phenomena related to phase coherence and localization are within reach with binary mixtures of BECs. Examples include the dynamics of component segregation and the study of the different dynamical regimes (see figure), some of which permit the indirect measurement of the spin dependent scattering length in spinor BECs. Other important aspects are related to the study of strongly correlated states that go beyond the mean-field; for those, our paper should serve as an important tool, incorporating all the existing mean-field approaches.