As the digital landscape evolves, Web3 has gained prominence, highlighting the critical role of decentralized, interconnected, and verifiable digital ecosystems. This paper introduces SPID-Chain, a novel interoperability consensus designed for Web3, which employs a directed acyclic graph (DAG) of blockchains to facilitate seamless integration across multiple blockchains. Within SPID-Chain, each blockchain maintains its own consensus and processes transactions via an intra-consensus mechanism that incorporates event-driven smart contracts (EDSC) and Polar codes for optimized computation distribution. This mechanism is complemented by a division of committee and worker nodes, enhancing transaction processing efficiency within individual chains. For inter-blockchain consensus, SPID-Chain utilizes a DAG structure where blockchains append blocks containing cross-chain transactions. These blocks are then processed through the inter-consensus mechanism orchestrated by the blockchains. Extensive simulations validate the efficacy of our scheme in terms of throughput, scalability, decentralization, and security. Our results showcase SPID-Chain's potential to enable fluid interactions and transactions across diverse blockchain networks, aligning with the foundational goals of Web3.