Magnetic ground states are commonly predicted using spin Hamiltonians whose interaction terms are selected a priori, potentially overlooking the microscopic interactions that govern complex magnetic order. Here, we introduce a general framework for the unbiased first-principles construction of symmetry-complete tensorial spin Hamiltonians and its automated implementation in AMATIS. The framework constructs the Hamiltonian directly from density-functional theory while rigorously enforcing quantum spin algebra and crystallographic symmetry. Applied to representative two-dimensional van der Waals magnets, the framework reproduces established magnetic interactions and uncovers hidden physics beyond conventional spin models, including chiral interactions that stabilize metastable skyrmions, higher-rank tensorial interactions that reconstruct the magnetic phase diagram and establish stabilizing competing multi-Q phases, and an emergent p-wave altermagnetic electronic structure. Our results demonstrate that unbiased tensorial Hamiltonian construction provides a predictive alternative to the conventional practice of manually selecting spin-model interactions, enabling first-principles discovery of unconventional magnetic phases.