This lecture explores the development of real Heegaard Floer homology as an invariant of based 3-manifolds with an involution. Discover the recent advancements in gauge theoretic invariants of 3- and 4-manifolds equipped with involutions, developed by researchers including Tian-Wang, Nakamura, Konno-Miyazawa-Taniguchi, and Li. Learn about the speaker's joint work with Ciprian Manolescu that constructs an analogue of Li's real monopole Floer homology. Understand how this construction represents a specialized case of real Lagrangian Floer homology, potentially valuable to symplectic geometers. Follow the development from a Heegaard diagram where involution swaps alpha and beta curves to proof that this represents a topological invariant of pointed real 3-manifolds. Examine the Euler characteristic of this theory as a Heegaard Floer analogue of Miyazawa's invariant for twist-spun 2-knots, noting its algorithmic computability and apparent agreement with Miyazawa's work.