A growth in data volume, combined with increasing demand for real-time analysis (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 losses in transactional (up to 74.6%) and/or analytical (up to 49.8%) throughput compared to performing only transactional or only analytical queries in isolation, due to (1) data movement between the CPU and memory, (2) data update propagation from transactional to analytical workloads, and (3) the cost to maintain a consistent view of data across the system. We propose Polynesia, a hardware-software co-designed system for in-memory HTAP databases that avoids the large throughput losses of traditional HTAP systems. Polynesia (1) divides the HTAP system into transactional and analytical processing islands, (2) implements new custom hardware that unlocks software optimizations to reduce the costs of update propagation and consistency, and (3) exploits processing-in-memory for the analytical islands to alleviate data movement overheads. Our evaluation shows that Polynesia outperforms three state-of-the-art HTAP systems, with average transactional/analytical throughput improvements of 1.7x/3.7x, and reduces energy consumption by 48% over the prior lowest-energy HTAP system.