Stacks of superconducting REBCO tapes (or “stacks”) can trap high magnetic fields, above 17 T. However, relatively low oscillating transverse magnetic fields can fully demagnetize the stacks. This is an issue if the stacks act as poles in the rotor of a superconducting motor, for instance. Here, we study the drastic suppression of cross-field demagnetization in stacks by soldering the tapes at the ends using a normal conductor. In particular, we analyzed by numerical modeling a stack of five REBCO thin films connected at the ends by resistances. The computed trapped field of a stack with zero solder resistance decays very fast at the beginning but then tends to stabilize to relatively high values, while the trapped field of an isolated stack (infinite resistance at the ends) decays further (it decays to zero if the transverse field is above the parallel penetration field). For intermediate solder resistances, the stable value of the trapped magnetic field is in between those of the isolated and zero-resistance configurations. Since the stable trapped field in soldered stacks increases with the number of tapes, stacks of sufficiently high number of soldered tapes could be immune to cross-field demagnetization. This opens the gate for a new kind of superconductors that mostly behave as bulks, especially if the stacks are made of delaminated tapes or it is possible to solder the tapes by very low resistance.