Tate and Artin--Tate Motivic Tensor-Triangular Geometry

We compute the Balmer spectra of geometric Tate and Artin--Tate motives with $\mathbb{F}_p$-coefficients over any field whose characteristic differs from $p$. For Tate motives, we find the bigraded comparison map to be a homeomorphism to the homogeneous spectrum of the motivic cohomology of the base