In a Minecraft Realm the option to turn on the coordinate system is not available without turning on cheats. That is not good because then you'll lose achievements.
Many tutorials recommend downloading the realm locally, flipping the switch for coordinates, and then uploading that copy of your world back to the realm. Turns out that some server commands will work in the Realm... coordinates being one of them!
Enter this into your chat window to enable coordinates:
/gamerule showcoordinates true
You can, of course, turn them back off via:
/gamerule showcoordinates false