Data outsourcing is a promising technical paradigm to facilitate cost-effective real-time data storage, processing, and dissemination. In such a system, a data owner proactively pushes a stream of data records to a third-party cloud server for storage, which in turn processes various types of queries from end users on the data owner's behalf. This paper considers outsourced multi-version key-value stores that have gained increasing popularity in recent years, where a critical security challenge is to ensure that the cloud server returns both authentic and fresh data in response to end users' queries. Despite several recent attempts on authenticating data freshness in outsourced key-value stores, they either incur excessively high communication cost or can only offer very limited real-time guarantee. To fill this gap, this paper introduces KV-Fresh, a novel freshness authentication scheme for outsourced key-value stores that offers strong real-time guarantee. KV-Fresh is designed based on a novel data structure, Linked Key Span Merkle Hash Tree, which enables highly efficient freshness proof by embedding chaining relationship among records generated at different time. Detailed simulation studies using a synthetic dataset generated from real data confirm the efficacy and efficiency of KV-Fresh.