I have. no experience with Eizo monitors, per se, but the general rule when cleaning the screen of any computer monitor is to first try with a wet microfiber cloth. That usually is all that's needed. Many glass-cleaning products and some specifically marketed as lens or screen cleaners can damage a coating on the monitor's surface.
To avoid leaving harmless but potentially visible streaks with the wet microfiber cloth technique, you may need to make a second pass over the surface with a dry microfiber cloth. Or you can use water that has been passed through a carafe with a filter manufactured by ZeroWater, which uses an ion-exchange technique to remove all the dissolved solids from tap water. (I have no connection with the company except as a retail customer.)
Always use a light touch when cleaning the screen of a computer monitor.