An exponential growth in data volume, combined with increasing demand for real-time analysis (i.e., using the most recent data), has resulted in the emergence of database systems that concurrently support transactions and data analytics. These hybrid transactional and analytical processing (HTAP) database systems can support real-time data analysis without the high costs of synchronizing across separate single-purpose databases. Unfortunately, for many applications that perform a high rate of data updates, state-of-the-art HTAP systems incur significant drops in transactional (up to 74.6%) and/or analytical (up to 49.8%) throughput compared to performing only transactions or only analytics in isolation, due to (1) data movement between the CPU and memory, (2) data update propagation, and (3) consistency costs. We propose Polynesia, a hardware-software co-designed system for in-memory HTAP databases. Polynesia (1) divides the HTAP system into transactional and analytical processing islands, (2) implements custom algorithms and hardware to reduce the costs of update propagation and consistency, and (3) exploits processing-in-memory for the analytical islands to alleviate data movement. Our evaluation shows that Polynesia outperforms three state-of-the-art HTAP systems, with average transactional/analytical throughput improvements of 1.70X/3.74X, and reduces energy consumption by 48% over the prior lowest-energy system.