Kinematically cold tidal streams of globular clusters (GC) are excellent tracers of the Galactic gravitational potential at moderate Galactocentric distances, and can also be used as probes of the law of gravity on Galactic scales. Here, we compare for the first time the generation of such streams in Newtonian and Milgromian gravity (MOND). We first computed analytical results to investigate the expected shape of the GC gravitational potential in both frameworks, and we then ran N-body simulations with the Phantom of Ramses code. We find that the GCs tend to become lopsided in MOND. This is a consequence of the external field effect which breaks the strong equivalence principle. When the GC is filling its tidal radius the lopsidedness generates a strongly asymmetric tidal stream. In Newtonian dynamics, such markedly asymmetric streams can in general only be the consequence of interactions with dark matter subhalos, giant molecular clouds, or interaction with the Galactic bar. In these Newtonian cases, the asymmetry is the consequence of a very large gap in the stream, whilst in MOND it is a true asymmetry. This should thus allow us in the future to distinguish these different scenarios by making deep observations of the environment of the asymmetric stellar stream of Palomar 5. Moreover, our simulations indicate that the high internal velocity dispersion of Palomar 5 for its small stellar mass would be natural in MOND.