Real syntomic cohomology

We introduce a theory of syntomic cohomology for ring spectra with involution, which we call Real syntomic cohomology. We show that our construction extends the theory of syntomic cohomology for rings with involution due to Park. Our construction also refines syntomic cohomology as developed by Bhatt–Morrow–Scholze, Morin, Bhatt–Lurie, and Hahn–Raksit–Wilson. We compute the Real syntomic cohomology of Real topological K-theory and topological modular forms with level structure.