A stable deformation $f^t$ of a real map-germ $f:ℝⁿ,0\to {ℝ}^p,0$ is said to be an M-deformation if all isolated stable (local and multi-local) singularities of its complexification $f_{ℂ}^t$ are real. A related notion is that of a good real perturbation $f^t$ of f (studied e.g. by Mond and his coworkers) for which the homology of the image (for n < p) or discriminant (for n ≥ p) of $f^t$ coincides with that of $f_C^t$. The class of map germs having an M-deformation is, in some sense, much larger than the one having a good real perturbation....