On integral aspects of Asai periods and Euler systems for $\mathrm{Res}_{E/\mathbf{Q}}\mathrm{GL}_2$

Let $E/\mathbf{Q}$ be a totally real quadratic field. Using unramified harmonic analysis in Hecke modules, we study the $\ell$-adic integral behavior of the (unramified part of the) Asai period attached to a Hilbert modular form for $E$, when evaluated on arbitrary integral test data in the sense of Loeffler. Using the same representation-theoretic framework, we also prove the conjectured integral behavior of local factors appearing in tame norm relations, between any collection of integral motivic Asai-Flach classes in the recipe of Loeffler-Skinner-Zerbes. Finally, specializing to one such specific integral collection, we obtain the most general version of the Asai-Flach Euler system tame norm relations, extending a result of Grossi.