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Can localization persist when interaction grows infinitely stronger than randomness? If so, is it many-body Anderson localization? How about the associated localization transition in the infinite-interaction limit? To tackle these questions, we study many-body localization (MBL) in a spin-chain model mimicking the Rydberg-blockade quantum simulator with both infinite-strength projection and moderate quasiperiodic modulation. Employing exact diagonalization, Krylov-typicality technique, and time-evolving block decimation, we identify evidence for a constrained MBL phase stabilized by a pure quasirandom transverse field. Remarkably, the constrained MBL transition may embody a discontinuous eigenstate phase transition, whose discontinuity nature significantly suppresses the finite-size drifts that plague most numerical studies of conventional MBL transition. Through quantum dynamics, we find that rotating the modulated field from parallel toward perpendicular to the projection axis induces an eigenstate transition between the diagonal and constrained MBL phases. Intriguingly, the entanglement-entropy growth in constrained MBL follows a double-log form, whereas it changes to a power law in approaching the diagonal limit. By unveiling the significance of confined nonlocal effects in integrals of motion of constrained MBL, we show that this newfound insulating state is not a many-body Anderson insulator. Our predictions can be tested in Rydberg experiments.