  We compute the contribution of order $\alpha_S^2\alpha^2$ to the cross
section of a top-antitop pair in association with at least one heavy Standard
Model boson -- $Z$, $W^\pm$, and Higgs -- by including all effects of QCD, QED,
and weak origin and by working in the automated MadGraph5_aMC@NLO framework.
This next-to-leading order contribution is then combined with that of order
$\alpha_S^3\alpha$, and with the two dominant lowest-order ones,
$\alpha_S^2\alpha$ and $\alpha_S\alpha^2$, to obtain phenomenological results
relevant to a 8, 13, and 100~TeV $pp$ collider.
