Plastics have substantial societal benefits, but their widespread use has led to a critical waste management challenge. While mechanical recycling dominates the reuse of post-consumer plastics, it is limited in efficacy, especially for composites. To address this, we propose a direct reprocessing approach that enables the creation of hybrid, long-lasting, and durable composites from difficult-to-recycle plastics. This approach utilizes cold sintering, a process that consolidates inorganic powders through fractional dissolution and precipitation at temperatures far below conventional sintering; these temperatures are compatible with plastic processing. We show that this process can create inorganic-matrix composites with significant enhancements in tensile strength and toughness over pure gypsum, which is commonly found in construction waste. These composites can be recycled multiple times through direct reprocessing with the addition of only water as a processing promoter. This approach to recycling leads to composites with orders of magnitude lower energy demand, global warming potential, and water demand, when compared against common construction products. Altogether, we demonstrate the potential for cold sintering to integrate waste into high-performance recyclable composites.