  We discuss some general properties of $\mathrm{TR}$ and its
$K(1)$-localization. We prove that after $K(1)$-localization, $\mathrm{TR}$ of
$H\mathbb{Z}$-algebras is a truncating invariant in the sense of Land--Tamme,
and deduce $h$-descent results. We show that for regular rings in mixed
characteristic, $\mathrm{TR}$ is asymptotically $K(1)$-local, extending results
of Hesselholt--Madsen. As an application of these methods and recent advances
in the theory of cyclotomic spectra, we construct an analog of Thomason's
spectral sequence relating $K(1)$-local $K$-theory and \'etale cohomology for
$K(1)$-local $\mathrm{TR}$.
