This paper presents a systematic approach to characterize the 't Hooft anomaly in open quantum systems, where symmetries can be classified as strong or weak due to their coupling with the environment. By representing symmetry transformations through superoperators, the authors derive a classification of anomalies for bosonic systems with symmetry group $ K \times G $, where $ K $ is the strong symmetry and $ G $ is the weak symmetry. Anomalies are classified by $ H^{d+2}(K \times G, U(1))/H^{d+2}(G, U(1)) $ in $ d $ spatial dimensions. The paper shows that anomalies in open quantum systems lead to nontrivial mixed-state quantum phases, similar to the "anomaly matching" condition in closed systems. A novel $ (1 + 1) $-D mixed-state quantum phase is identified, where the steady state exhibits spontaneous symmetry breaking on the boundary, enforced by anomalies. The authors also generalize the "anomaly inflow" mechanism to open quantum systems, constructing $ (1 + 1) $-D and $ (2 + 1) $-D Lindbladians with mixed-state symmetry-protected topological order in the bulk, characterized by nontrivial anomalies. The paper demonstrates that anomalies in open quantum systems constrain mixed-state quantum phases, leading to nontrivial steady states and long-time dynamics. The results highlight the importance of anomalies in understanding mixed-state quantum phases and their implications for open quantum systems.This paper presents a systematic approach to characterize the 't Hooft anomaly in open quantum systems, where symmetries can be classified as strong or weak due to their coupling with the environment. By representing symmetry transformations through superoperators, the authors derive a classification of anomalies for bosonic systems with symmetry group $ K \times G $, where $ K $ is the strong symmetry and $ G $ is the weak symmetry. Anomalies are classified by $ H^{d+2}(K \times G, U(1))/H^{d+2}(G, U(1)) $ in $ d $ spatial dimensions. The paper shows that anomalies in open quantum systems lead to nontrivial mixed-state quantum phases, similar to the "anomaly matching" condition in closed systems. A novel $ (1 + 1) $-D mixed-state quantum phase is identified, where the steady state exhibits spontaneous symmetry breaking on the boundary, enforced by anomalies. The authors also generalize the "anomaly inflow" mechanism to open quantum systems, constructing $ (1 + 1) $-D and $ (2 + 1) $-D Lindbladians with mixed-state symmetry-protected topological order in the bulk, characterized by nontrivial anomalies. The paper demonstrates that anomalies in open quantum systems constrain mixed-state quantum phases, leading to nontrivial steady states and long-time dynamics. The results highlight the importance of anomalies in understanding mixed-state quantum phases and their implications for open quantum systems.