Unwinding the relative Tate diagonal

We show that a spectral sequence developed by Lipshitz and Treumann, for application to Heegaard Floer theory, converges to a localized form of topological Hochschild homology with coefficients. This allows us to show that the target of this spectral sequence can be identified with Hochschild homology when the topological Hochschild homology is torsion-free as a module over $\mathrm{THH}_*(\mathbb{F}_2)$, parallel to results of Mathew on degeneration of the Hodge-to-de Rham spectral sequence. To carry this out, we apply work of Nikolaus-Scholze to develop a general Tate diagonal for Hochschild-like diagrams of spectra that respect a decomposition into tensor products. This allows us to discuss the extent to which there can be a Tate diagonal for relative topological Hochschild homology.