After Leffingwell kills Tattoo and returns to Choya, he has Tattoo's horse. Choya even mentions it. But in the next scene, the extra horse is nowhere to be seen.
When Alan Ladd's character is washing up (takes his shirt off to reveal the fake birthmark) after riding the unbroken horse, he uses a faucet from a modern pressurized water system rather than a period hand pump.