  This paper presents a combined analysis of the potential of a future
electron-positron collider to constrain the Higgs, top and electro-weak (EW)
sectors of the Standard Model Effective Field Theory (SMEFT). The leading
contributions of operators involving top quarks arise mostly at one-loop
suppressed order and can be captured by the renormalization group mixing with
Higgs operators. We perform global fits with an extended basis of 29
parameters, including both Higgs and top operators, to the projections for the
Higgs, top and electro-weak precision measurements at the International Linear
Collider (ILC). The determination of the Higgs boson couplings in the 250 GeV
stage of the ILC is initially severely degraded by the additional top-quark
degrees of freedom, but can be nearly completely recovered by the inclusion of
precise measurements of top-quark EW couplings at the LHC. The physical Higgs
couplings are relatively robust, as the top mass is larger than the energy
scale of EW processes. The effect of the top operators on the bounds on the
Wilson coefficients is much more pronounced and may limit our ability to
identify the source of deviations from the Standard Model. Robust global bounds
on all Wilson coefficients are only obtained when the 500 GeV stage of the ILC
is included.
