I own a halter-bridle but have not used it in years because I don’t care for the way the cheekpieces attach to the main headstall, and it’s a headache to change bits. You practically have to take the entire cheekpiece apart.
There are some halter-bridles on the market where the bit attaches to a single strap of leather/biothane material, which snaps into the headstall/halter in one or two places. I think Brenda Imus’ halter/bridle works that way, though I’ve had bad experiences with one of her other products and found her customer service to be very lacking, so she has completely lost me as a customer now and in the future. There are other halter-bridles around. . .I think windrider has a similar style made out of biothane and I’ve considered ordering one.
For the moment, I ride in a leather English-style bridle and either pack along one of those wide nylon horse-collars with a lead rope, or he wears a rope halter under the bridle. The other TWH I’ve been riding goes in an English bridle as well, with a cavesson, and a rope halter beneath it. It all depends on how much of a ride I plan on doing. If it’s only 3-4 hours, I “go light” and may not even bring a halter or head-collar to tie the horse.