The motivic tt-geometry of real quadrics

We study the tensor-triangular geometry of the category of Voevodsky motives generated by real quadrics. At the prime 2, we determine its Balmer spectrum, and find that it is a countably infinite, non-Noetherian space of Krull dimension 2. We detail the relationship between this space, the real Artin-Tate spectrum computed by Balmer-Gallauer, and Vishik’s isotropic points. We conclude by combining our computation with Balmer-Gallauer’s results on Artin-Tate motives to obtain a full description of the spectrum of integral motives of quadrics over real algebraic numbers.