  Non-decoupling D-term extensions of the MSSM enhance the tree-level Higgs
mass compared to the MSSM, therefore relax fine-tuning and may allow lighter
stops with rather low masses even without maximal mixing. We present the
anatomy of various non-decoupling D-term extensions of the MSSM and explore the
potential of the LHC and of the International Linear Collider (ILC) to
determine their deviations in the Higgs couplings with respect to the Standard
Model. Depending on the mass of the heavier Higgs $m_H$, such deviations may be
constrained at the LHC and determined at the ILC. We evaluate the Higgs
couplings in different models and study the prospects for a model distinction
at the different stages of the ILC at $\sqrt{s}=$250, 500, 1000 GeV, including
the full luminosity upgrade and compare it with the prospects at HL-LHC.
