We report angle-and momentum-resolved measurements of the dissociative ionization and Coulomb explosion of methyl halides (CH 3 F, CH 3 Cl, CH 3 Br, CH 3 I) in intense phase-controlled two-color laser fields. At moderate laser intensities we find that the emission asymmetry of low-energy CH + 3 fragments from the CH + 3 + X + (X = F,Cl,Br,I) channel reflects the asymmetry of the highest occupied molecular orbital of the neutral molecule with important contributions from the Stark effect. This asymmetry is correctly predicted by the weak-field asymptotic theory, provided that the Stark effect on the ionization potentials is calculated using a non-perturbative multi-electron approach. In the case of high laser intensities we observe a reversal of the emission asymmetries for high-energy CH + 3 fragments, originating from the dissociation of CH 3 X q+ with q ≥ 2. We propose ionization to electronically-excited states to be at the origin of the reversed asymmetries. We also report the measurements of the emission asymmetry of H + 3 which is found to be identical to that of the low-energy CH + 3 fragments measured at moderate laser intensities. All observed fragmentation channels are assigned with the help of CCSD(T) calculations. Our results provide a benchmark for theories of strong-field processes and demonstrate the importance of multi-electron effects in new aspects of the molecular response to intense laser fields.