Self-consistent simulations of the interaction of initially spherical low-mass satellite galaxies with a massive Galactic dark halo suggest that observed apparent high mass-to-light ratios, ( M/L) obs, in dwarf-spheroidal (dSph) satellites may be obtained through successive and destructive perigalactic passages. The model satellites are disrupted after several perigalactic passages, but orbiting condensations in phase space can clearly be identified as remnants well after this event. An intrinsic ( M/L) true < 3 leads to long-lived model remnants that show, as seen from Earth, a tantalizingly close resemblance to observed dSph satellites with ( M/L) obs > 10 . The reason for large ( M/L) obs is that the remnants have non-isotropic velocity dispersions, are non-spherical and are not in dynamical equilibrium. The remnants fade with time (i.e. orbital phase) and ( M/L) obs increases indicating a correlation between brightness and ( M/L) obs as in the observed dSph sample. The observed correlation between central surface brightness and integrated absolute magnitude is also approximately reproduced by model remnants of different dynamical age. A possibly useful method for probing the line-of-sight extension of a dSph galaxy is suggested. If it is true that most if not all dSph satellites are tidally modified remnants without dark matter then it is plausible that their progenitors may have formed as satellite galaxies in tidal tails during possible early merging events.