K-theoretic Tate-Poitou duality at prime 2

We extend the result of Blumberg and Mandell on K-theoretic Tate-Poitou duality at odd primes which serves as a spectral refinement of the classical arithmetic Tate-Poitou duality. The duality is formulated for the $K(1)$-localized algebraic K-theory of the ring of $p$-integers in a number field and its completion using the $\mathbb{Z}_p$-Anderson duality. This paper completes the picture by addressing the prime 2, where the real embeddings of number fields introduce extra complexities. As an application, we identify the homotopy type at prime 2 of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic K-theory of the integers.