  We study the equilibrium properties of a one-dimensional mixture of two
Tonks-Girardeau gases on a ring geometry in the limit of strongly-repulsive
inter-species interactions. We derive the exact many-body wavefunction and
compare it to the $SU(2)$ solution where intra- and inter-species interactions
are also diverging but equal. We focus on the role of the $SU(2)$-symmetry
breaking on the behaviour of the large- and short-distance correlations by
studying the zero-momentum occupation number and the Tan's contact from the
asymptotic behavior of the momentum distribution. Although the symmetry is only
weakly broken, it has important consequences on spin correlations in the system
as the reduction by a factor of two of the zero-momentum occupation number with
respect to the $SU(2)$ case in the thermodynamic limit and the decrease of the
Tan's contact.
