We simulate anisotropic thermal conduction between the intracluster medium (ICM) and the hot coronal interstellar medium (ISM) gas in cluster galaxies. In the earlier Paper I (Vijayaraghavan & Sarazin 2017a), we simulated the evaporation of the hot ISM due to isotropic (possibly saturated) conduction between the ISM and ICM. We found that hot coronae evaporate on $sim 10^2$ Myr timescales, significantly shorter than the $sim 10^3$ Myr gas loss times due to ram pressure stripping. No tails of stripped gas are formed. This is in tension with the observed ubiquity and implied longevity of compact X-ray coronae and stripped ISM tails, and requires the suppression of evaporation, possibly due to magnetic fields and anisotropic conduction. We perform a series of wind tunnel simulations similar to Paper I, now including ISM and ICM magnetic fields. We simulate the effect of anisotropic conduction for a range of extreme magnetic field configurations: parallel and perpendicular to the ICM wind, and continuous and completely disjoint between the ISM and ICM. We find that when conduction is anisotropic, gas loss due to evaporation is severely reduced; the overall gas loss rates with and without anisotropic conduction do not differ by more than $10 - 20%$. Magnetic fields also prevent stripped tails from evaporating in the ICM by shielding, and providing few pathways for heat transport between the ICM and ISM. The morphology of stripped tails and magnetic fields in the tails and wakes of galaxies are sensitive to the initial magnetic field configuration.
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