Gamma-ray bursts (GRBs) are powered by relativistic jets that exhibit intermittency over a broad range of timescales - from $ sim $ ms to seconds. Previous numerical studies have shown that hydrodynamic (i.e., unmagnetized) jets that are expelled from a variable engine are subject to strong mixing of jet and cocoon material, which strongly inhibits the GRB emission. In this paper we conduct 3D RMHD simulations of mildly magnetized jets with power modulation over durations of 0.1 s and 1 s, and a steady magnetic field at injection. We find that when the jet magnetization at the launching site is $sigma sim 0.1$, the initial magnetization is amplified by shocks formed in the flow to the point where it strongly suppresses baryon loading. We estimate that a significant contamination can be avoided if the magnetic energy at injection constitutes at least a few percent of the jet energy. The variability timescales of the jet after it breaks out of the star are then governed by the injection cycles rather than by the mixing process, suggesting that in practice jet injection should fluctuate on timescales as short as $ sim 10 $ ms in order to account for the observed light curves. Better stability is found for jets with shorter modulations. We conclude that for sufficiently hot jets, the Lorentz factor near the photosphere can be high enough to allow efficient photospheric emission. Our results imply that jets with $ 10^{-2} < sigma < 1 $ injected by a variable engine with $ sim 10 $ ms duty cycle are plausible sources of long GRBs.
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