  We address the problem of calculating the correlation functions of
one-dimensional two-component gases with strong repulsive contact interactions.
The model considered in this paper describes particles with fractional
statistics and in appropriate limits reduces to the Gaudin-Yang model or the
spinor Bose gas. In the case of impenetrable particles we derive a Fredholm
determinant representation for the temperature-, time-, and space-dependent
correlation functions which is very easy to implement numerically and
constitute the starting point for the analytical investigation of the
asymptotics. Making use of this determinant representation and the solution of
an associated Riemann-Hilbert problem we derive the low-energy asymptotics of
the correlators in the spin-incoherent regime characterized by near
ground-state charge degrees of freedom but a highly thermally disordered spin
sector. The asymptotics present features reminiscent of spin-charge separation
with the spin part exponentially decaying in space separation and oscillating
with a period proportional to the statistics parameter while the charge part
presents scaling with anomalous exponents which cannot be described by any
unitary conformal field theory. The momentum distribution and the Fourier
transform of the dynamical Green's function are asymmetrical for arbitrary
statistics, a direct consequence of the broken space-reversal symmetry. Due to
the exponential decay the momentum distribution $n(k)$ at zero temperature does
not present algebraic singularities but the tails obey the universal decay
$\lim_{k\rightarrow\pm\infty}n(k)\sim C/k^4$ with the amplitude $C$ given by
Tan's contact. As a function of the statistics parameter the contact is a
monotonic function reaching its minimum for the fermionic system and the
maximum for the bosonic system.
