We report angle- and momentum-resolved measurements of the dissociative ionization and Coulomb explosion of methyl halides (CH3F, CH3Cl, CH3Br, and CH3I) in intense phase-controlled two-color laser fields. At moderate laser intensities, we find that the emission asymmetry of low-energy CH3+ fragments from the CH3+ + X+ (X = F, Cl, Br, or 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 nonperturbative multielectron approach. In the case of high laser intensities, we observe a reversal of the emission asymmetries for high-energy CH3+ fragments, originating from the dissociation of CH3Xq+ 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 CH3+ 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 multielectron effects in new aspects of the molecular response to intense laser fields.