The players recently acquired a figurine of wondrous power (elephant) and are considering using it as a combat platform in an upcoming boss fight (rather than just commanding it to attack).
I noted that the elephant does not specifically come equipped with a saddle / howdah and began to look for what sort of consequences would apply to riding it without a saddle. To my surprise, there don't appear to be any.
The PHB (Mounts and Vehicles) section says:
Saddles. A military saddle braces the rider, helping you keep your seat on an active mount in battle. It gives you advantage on any check you make to remain mounted. An exotic saddle is required for riding any aquatic or flying mount.
Although interpretations of this could vary, I think the most straightforward conclusion is that while a military saddle gives you advantage on checks to remain mounted, and an exotic saddle is required to ride a mount that is aquatic or flying, there is no RAW effect of a normal (riding) saddle, and no consequences for riding a terrestrial mount without one.
The PHB section on Mounted Combat describes mounting, dismounting, and what happens to you if your mount is moved against its will or knocked prone. None of these rules reference interacting with saddles, however.
Similarly, in real life a Bridle is used by a rider to guide a horse in the desired direction (the bridle is what the reins are attached to, although reins are not offered in the PHB Tack and Harness equipment list). A difficult-to-control mount might also, in real life, have a Bit, which applies pressure to the sensitive tissue of the mouth, but these are not required for well-trained or responsive mounts. However, as far as I can tell, there are not any RAW effects for a Bit and Bridle either, having checked the PHB section on "Controlling a Mount".
Thus, I am asking whether I have missed something – Are there any RAW effects of a Riding Saddle, Bit, or Bridle? Conversely, are there any RAW effects of riding a mount (such as the elephant) without these items?
I am not asking for suggestions on how to apply circumstantial effects based on the DM's power to adjudicate – I am capable of doing that myself.
An answer could argue that I am reading the PHB incorrectly, or that there are mounted combat rules tucked away in the DMG that I have not spotted.
A good answer might cite optional rules in some other official product of which I am unaware.
A humorous answer might cite an Adventurer's League situation in which a player had refused to purchase saddle, bit, and bridle for their mount to save weight and money and the DM was forced to conclude that there was no mechanical difference for them.
Best Answer
There are no RAW effects of these items other than the ones you have found. You have found the complete set of rules regarding mounted combat. There are no others in published D&D material.
To confirm this, I searched D&D Beyond for saddle, bit, and bridle. None of those turn up any result for rules beyond the ones you have found. Some turn up magical items of these sorts -- the saddle of the cavalier in particular -- but none of those are rules for saddles, bits, or bridles in general.
(Personally, I'm of the opinion that rules penalizing riders without a bit, bridle, or saddle would almost exclusively punish cool scenes in which players jump on wild animals or hostile beasts and try to gain control of them, and I'm pleased such rules do not exist.)