Motivic real topological Hochschild spectrum

We define real topological Hochschild homology of separated log schemes with involutions. We show that real topological Hochschild homology is $({\mathbb{P}}^n,{\mathbb{P}}^{n-1})$-invariant, which leads to the definition of the motivic real topological Hochschild spectrum living in a certain $\mathbb{Z}\mathrm{/}2$-equivariant logarithmic motivic category. We explore properties of real topological Hochschild homology that can be deduced from the logarithmic motivic homotopy theory. We also define the motivic real topological cyclic spectrum.