Product search has been a crucial entry point to serve people shopping online. Most existing personalized product models follow the paradigm of representing and matching user intents and items in the semantic space, where finer-grained matching is totally discarded and the ranking of an item cannot be explained further than just user/item level similarity. In addition, while some models in existing studies have created dynamic user representations based on search context, their representations for items are static across all search sessions. This makes every piece of information about the item always equally important in representing the item during matching with various user intents. Aware of the above limitations, we propose a review-based transformer model (RTM) for personalized product search, which encodes the sequence of query, user reviews, and item reviews with a transformer architecture. RTM conducts review-level matching between the user and item, where each review has a dynamic effect according to the context in the sequence. This makes it possible to identify useful reviews to explain the scoring. Experimental results show that RTM significantly outperforms state-of-the-art personalized product search baselines.