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We recently developed a new general relativistic magnetohydrodynamic code with adaptive mesh refinement that evolves the electromagnetic (EM) vector potential A, instead of the magnetic fields directly. Evolving A; enables one to use any interpolation scheme on refinement level boundaries and still guarantee that the magnetic field remains divergenceless. As in classical EM, a gauge choice must be made when evolving A,, and we chose a straightforward "algebraic" gauge condition to simplify the A, evolution equation. However, magnetized black hole-neutron star (BHNS) simulations in this gauge exhibit unphysical behavior, including the spurious appearance of strong magnetic fields on refinement level boundaries. This spurious behavior is exacerbated when matter crosses refinement boundaries during tidal disruption of the neutron star. Applying Kreiss-Oliger dissipation to the evolution of the magnetic vector potential A, slightly weakens this spurious magnetic effect, but with undesired consequences. We demonstrate via an eigenvalue analysis and a numerical study that zero-speed modes in the algebraic-gauge, coupled with the frequency filtering that occurs on refinement level boundaries, are responsible for the creation of spurious magnetic fields. We show that the EM Lorenz gauge exhibits no zero-speed modes, and as a consequence, spurious magnetic effects are quickly propagated away, allowing for long-term, stable magnetized BHNS evolutions. Our study demonstrates how the EM gauge degree of freedom can be chosen to one's advantage, and that for magnetized BHNS simulations the Lorenz gauge constitutes a major improvement over the algebraic gauge. [ABSTRACT FROM AUTHOR]