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The Tomonaga-Luttinger liquid (TLL) theory describes the low-energy excitations of strongly correlated one-dimensional (1D) fermions. In the past years, a number of studies have provided a detailed understanding of this universality class. More recently, theoretical investigations that go beyond the standard low-temperature, linear-response, TLL regime have been developed. While these provide a basis for understanding the dynamics of the spin-incoherent Luttinger liquid, there are few experimental investigations in this regime. Here we report the observation of a thermally-induced, spin-incoherent Luttinger liquid in a $^6$Li atomic Fermi gas confined to 1D. We use Bragg spectroscopy to measure the suppression of spin-charge separation and the decay of correlations as the temperature is increased. Our results probe the crossover between the coherent and incoherent regimes of the Luttinger liquid, and elucidate the roles of the charge and the spin degrees of freedom in this regime.