  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
$\mathbb{Z}/2$-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.
