No one size fits all answer It depends on how the artificial patina was applied. Maybe water, maybe solvents such as acetone or xylene, maybe something else will remove it.
Successful removal of an artificial patina depends on what was applied. I've had coins that I'm pretty certain had women's makeup applied (a sandy color). Isopropyl alcohol worked for those. Colored waxes may require mineral spirits. Acetone is often helpful in removing applied patinas.
As has been stated previously, patina, artificial or natural takes study of the compound and prior trial and error to affect. Some coins have a very hard, glossy, and fragile patina which is often some sort of cyanoacrylate (super glue); I've seen what I believe to be super glued sand patinas. In this case, acetone works well. Sometimes you have a Renaissance wax patina; these can be removed by acetone or alcohol, usually. Sometimes you have a rather dense, paraffin wax-like situation in which case boiling might actually work, though I've never tried it. Chemically-applied patinas, such as Jax Patina Restorer is hard to remove and I've yet to find a good way to do so. This is a liquid that will seep into every pore and sort of bind with the metal; removal of this can often damage the rest of the patina. Electrolysis is a good way of stripping a coin to bare metal - which allows one to repatinate how they please. Finally, a rust remover such as Naval Jelly will absolutely get absolutely everything off the surface of a coin. Danger is that it goes too far and starts to dissolve the metal itself. I've tried this before and ended up with some chemical burns and coins that pretty much lost all meaning. Not recommended, but it will "work"
Going back to the very original spirit of this thread, I have several coins I suspect to have an applied sand patina. However, the sand doesn't cover the entire fields, and they look pretty, so I leave them alone. Example below
You would like them better painted pink and green. Attractiveness is part of the con when they artificially treat the surface like that.
Not really, just going with pure facts. The best way to prepare a coin for artificial patination is electrolysis, it's simple as that. Just like the best way to paint a car is to sand/strip it down to bare metal, same goes for coins.
The beautifully crisp and clear inscriptions would likely suffer if you were to attempt to remove the patina. I would leave them alone. Here’s a link to a 2017 CoinTalk discussion about desert patina that you may find interesting. https://www.cointalk.com/threads/how-do-i-spot-fake-desert-patina.305141/