Conversational Product Search ( CPS ) systems interact with users via natural language to offer personalized and context-aware product lists. However, most existing research on CPS is limited to simulated conversations, due to the lack of a real CPS dataset driven by human-like language. Moreover, existing conversational datasets for e-commerce are constructed for a particular market or a particular language and thus can not support cross-market and multi-lingual usage. In this paper, we propose a CPS data collection protocol and create a new CPS dataset, called PSCon, which assists product search through conversations with human-like language. The dataset is collected by a coached human-human data collection protocol and is available for dual markets and two languages. By formulating the task of CPS, the dataset allows for comprehensive and in-depth research on six subtasks: user intent detection, keyword extraction, system action prediction, question selection, item ranking, and response generation. Moreover, we present a concise analysis of the dataset and propose a benchmark model on the proposed CPS dataset. Our proposed dataset and model will be helpful for facilitating future research on CPS.