  We derive the equations for the non-linear effective dynamics of a so called
pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body
Schr\"odinger equation at the leading order in the number of particles. The
considered system is a three-dimensional diluted gas of identical bosons with
spin, possibly confined in space, and coupled with an external time-dependent
magnetic field; particles also interact among themselves through a short-scale
repulsive interaction. The limit of infinitely many particles is monitored in
the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at
time zero the system is in a phase of complete condensation (at the level of
the reduced one-body marginal) and with energy per particle fixed by the
Gross-Pitaevskii functional, then such conditions persist also at later times,
with the one-body orbital of the condensate evolving according to a system of
non-linear cubic Schr\"odinger equations coupled among themselves through
linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of
Pickl's projection counting method developed for the scalar case. Quantitative
rates of convergence are available, but not made explicit because evidently
non-optimal. In order to substantiate the formalism and the assumptions made in
the main theorem, in an introductory section we review the mathematical
formalisation of modern typical experiments with pseudo-spinor condensates.
