  We present a complete description of top quark pair production in association
with a hard photon in the dilepton channel. Our calculation is accurate to NLO
in QCD. It is based on matrix elements for $e^+\nu_e \mu^-\bar{\nu}_\mu b
\bar{b}\gamma$ production and includes all resonant and non-resonant diagrams,
interferences, and off-shell effects of the top quarks and the $W$ gauge
bosons. This calculation constitutes the first full computation for top quark
pair production with a final state photon in hadronic collisions at NLO in QCD.
Numerical results for total and differential cross sections are presented for
the LHC at a centre-of-mass energy of $\sqrt{s}=13$ TeV. For a few observables
relevant for new physics searches, beyond some kinematic bounds, we observe
shape distortions of more than $100\%$. In addition, we confirm that the size
of the top quark off-shell effects for the total cross section is consistent
with the expected uncertainties of the narrow width approximation. Results
presented here are not only relevant for beyond the Standard Model physics
searches but also important for precise measurements of the top-quark fiducial
cross sections and top-quark properties at the LHC.
