Unconventional superconductivity under strong Zeeman fields—particularly beyond the Pauli paramagnetic limit—remains a central challenge in condensed matter physics. The exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in particular, remains in need of definitive study within fundamental electronic models. Here we employ state-of-the-art finite-temperature and ground-state tensor network approaches to systematically explore the superconducting (SC) phase diagram of the $t\text{−}{t}^{\prime }\text{−}J$ model subjected to Zeeman fields. We find that zero-momentum $d$-wave superconductivity persists until the spin gap closes, coexisting with charge density waves. A novel $d$-wave FFLO phase emerges under a higher Zeeman field even above the Pauli limit, concomitant with a field-enhanced spin density waves. We identify these phases, characterized by the simultaneous presence of pairing condensate and density wave orders, as charge-2e supersolids. Analysis of Matsubara Green's function reveals that the FFLO pairing momentum is locked to the underlying Fermi surface. Our results provide microscopic insights into field-induced unconventional pairing mechanisms and reveal the long-sought FFLO state in a fundamental correlated electron model, offering a promising route for its realization in ultracold atom optical lattice.

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• Cold atoms & matter waves
• FFLO
• Strongly correlated systems
• Tensor network methods
• t-J model