The proliferation of multi-agent systems (MAS) in complex, dynamic environments necessitates robust and efficient mechanisms for managing shared knowledge. A critical challenge is ensuring that distributed memories remain synchronized, relevant, and free from the accumulation of outdated or inconsequential data - a process analogous to biological forgetting. This paper introduces the Co-Forgetting Protocol, a novel, comprehensive framework designed to address this challenge by enabling synchronized memory pruning in MAS. The protocol integrates three key components: (1) context-aware semantic voting, where agents utilize a lightweight DistilBERT model to assess the relevance of memory items based on their content and the current operational context; (2) multi-scale temporal decay functions, which assign diminishing importance to memories based on their age and access frequency across different time horizons; and (3) a Practical Byzantine Fault Tolerance (PBFT)-based consensus mechanism, ensuring that decisions to retain or discard memory items are agreed upon by a qualified and fault-tolerant majority of agents, even in the presence of up to f Byzantine (malicious or faulty) agents in a system of N greater than or equal to 3f+1 agents. The protocol leverages gRPC for efficient inter-agent communication and Pinecone for scalable vector embedding storage and similarity search, with SQLite managing metadata. Experimental evaluations in a simulated MAS environment with four agents demonstrate the protocol's efficacy, achieving a 52% reduction in memory footprint over 500 epochs, 88% voting accuracy in forgetting decisions against human-annotated benchmarks, a 92% PBFT consensus success rate under simulated Byzantine conditions, and an 82% cache hit rate for memory access.