  We define the motivic filtrations on real topological Hochschild homology and
its companions. In particular, we prove that real topological cyclic homology
admits a natural complete filtration whose graded pieces are equivariant
suspensions of syntomic cohomology. As an application, we compute the
equivariant slices of $p$-completed real $K$-theories of $\mathbb{F}_p[x]/x^e$
and $\mathbb{Z}/p^n$ after certain suspensions assuming a real refinement of
the Dundas-McCarthy-Goodwillie theorem and an announced result of
Antieau-Krause-Nikolaus.
