As the concrete industry moves toward sustainable, automated construction, understanding the rheological behavior of alternative binders is essential. In particular, the rheology of limestone calcined clay cement (LC3) is extremely sensitive to the type of calcium sulfate used. This study systematically investigates the impact of anhydrite (CaSO4), bassanite (CaSO4⋅0.5 H2O), and gypsum (CaSO4⋅2 H2O) on hydration kinetics, structural build-up, and workability of LC³ pastes. Isothermal calorimetry, rotational and oscillatory rheometry (Large-Amplitude Oscillatory Shear (LAOS) tests) were used to decouple the interplays between sulfate dissolution, hydration and thixotropic behavior. The results indicate that bassanite accelerates early-age structuration due to its rapid dissolution and ettringite formation, yielding a high structuration rate (Athix = 0.5 Pa/min) and optimal shear stress evolution (up to 102 Pa). Conversely, gypsum retards structuration and extends workability beyond 140 minutes, but compromises early stiffening. Anhydrite, despite its coarser morphology, exhibited intermediate behavior with rapid workability reduction. LAOS analysis also identified distinct viscoelastic thresholds. Pastes with bassanite reached critical strain (10^-3) and crossover strain (10^-2) at minimal deformation, ideal for automated construction, while gypsum formulations showed delayed stiffening. This study demonstrates that sulfate selection directly controls open time, with bassanite formulations requiring a 90-minute operational time frame to balance extrudability and layer stability. These findings underscore the need to tailor calcium sulfate type to application-specific rheological demands and offer a pathway to optimize LC3 binders for automated processes such as robotic shotcreting and 3D concrete printing.