AI-augmented data processing systems (DPSs) integrate large language models (LLMs) into query pipelines, allowing powerful semantic operations on structured and unstructured data. However, the reliability (a.k.a. trust) of these systems is fundamentally challenged by the potential for LLMs to produce errors, limiting their adoption in critical domains. To help address this reliability bottleneck, we introduce semantic integrity constraints (SICs) —a declarative abstraction for specifying and enforcing correctness conditions over LLM outputs in semantic queries. SICs generalize traditional database integrity constraints to semantic settings, supporting common types of constraints, such as grounding, soundness, and exclusion, with both reactive and proactive enforcement strategies. We argue that SICs provide a foundation for building reliable and auditable AI-augmented data systems. Specifically, we present a system design for integrating SICs into query planning and runtime execution and discuss its realization in AI-augmented DPSs. To guide and evaluate our vision, we outline several design goals—covering criteria around expressiveness, runtime semantics, integration, performance, and enterprise-scale applicability—and discuss how our framework addresses each, along with open research challenges.