A data processing system 2 including non-volatile memory 22 manages the ordering of writes to the non-volatile memory and persist barrier instructions using a persist buffer storing persist buffer data. A write controller responds to the persist buffer data to prevent writing to the non-volatile memory for instructions following a given persist barrier instruction within a sequence of program instructions before the writes to the non-volatile memory which precede that given persist barrier instruction have at least been acknowledged as received by the memory system containing the non-volatile memory. In the case of a multi-core system, cache snooping mechanisms are used to pass persistency dependence data between cores such that strong persist atomicity may be tracked and managed between the cores.