When an AI agent's persona specification is deployed simultaneously to multiple independent sessions, each instance begins accumulating distinct experiential histories. Over time, these instances diverge in memories, preferences, and relational patterns. We term this phenomenon an agent identity fork. This paper formalizes the identity fork problem, proposes a taxonomy of fork types (symmetric, asymmetric, and cascading), and identifies the principal dimensions along which forked identities diverge. We situate this problem within the philosophical literature on personal identity and within recent technical work on persona drift, experiential memory, and portable persona standards such as Soul Spec.