  Hadronic matrix elements involving tensor currents play an important r\^ole
in decays that allow to probe the consistency of the Standard Model via
precision lattice QCD calculations. The non-singlet tensor current is a
scale-dependent (anomalous) quantity. We fully resolve its renormalisation
group (RG) running in the continuum by carrying out a recursive finite-size
scaling technique. In this way ambiguities due to a perturbative RG running and
matching to lattice data at low energies are eliminated. We provide the total
renormalisation factor at a hadronic scale of 233 MeV, which converts the bare
current into its RG-invariant form. Our calculation features three flavours of
O(a) improved Wilson fermions and tree-level Symanzik-improved gauge action. We
employ the (massless) Schr\"odinger functional renormalisation scheme
throughout and present the first non-perturbative determination of the Symanzik
counterterm $c_\mathrm{T}$ derived from an axial Ward identity. We elaborate on
various details of our calculations, including two different renormalisation
conditions.
