Forgot to mention, disconnecting and re-connecting to the realm fixes that instance of the glitch. It's easy for my PC, but for others it can take some time to rejoin.
I'm having this issue on realms, seems to be client based. I dis-mount my horse and it visually disappears but I can still hear it and my cousin (who is playing on the same realm) can see the horse just fine.
Tagnan Ribaudo
Forgot to mention, disconnecting and re-connecting to the realm fixes that instance of the glitch. It's easy for my PC, but for others it can take some time to rejoin.