The Pristine survey XXXI. A stream offset from the Stream: very metal-poor stellar debris trailing 100$^{\circ}$ from the LMC-SMC system
Context: The stellar counterpart to the $200^\circ$-long HI Stream from the Clouds probes their tidally stripped stellar populations and constrains their interaction history and structure before infall into the Milky Way. Aims: We investigate the chemodynamics and possible origin of very metal-poor (VMP; [Fe/H] $<-2$) stars in the stellar Stream, targeting distant red giants selected from the Pristine--Gaia synthetic metallicity catalogue. Methods: We obtained high-resolution Gemini/GHOST spectroscopy of 33 candidate members, deriving radial velocities and metallicities. We combine these data with literature red giant members and assess their association with the Clouds using orbital integrations in a time-dependent Milky Way--LMC potential. We also compare their distribution with collisionless test particles initially placed in a diffuse LMC stellar halo. Results: We identify 12 high-confidence members based on energy and angular momentum: one intermediate metal-poor star ([Fe/H] $\sim-1.6$), ten VMP stars, and one extremely metal-poor (EMP) star ([Fe/H] $<-3$). The new VMP stars trace over $100^\circ$ of the trailing Stream, follow the Clouds' kinematics, and lie $\sim20^\circ$ from both the HI Stream and previously known, more metal-rich debris, suggesting a metallicity-dependent spatial distribution. Combined with the literature, they approximately double the known red giant sample and extend its metallicity range from [Fe/H] $\sim-0.5$ to $-4.3$. Orbital integrations support an association with the Clouds, while test particles reproduce the debris' broad spatial distribution. Conclusions: VMP and EMP stars contribute to the Clouds' tidal debris. They may originate in the extended stellar halos of the LMC or SMC, or in disrupted satellites. Their spatial and kinematic properties can constrain future simulations of the Milky Way--LMC--SMC interaction.