The only time I'll mix ammo is during a hike in snake territory. The first chamber in the cylinder is loaded with snake shot, the rest with FMJ. For CC, I've always done the same ammunition type (FMJ in a 1911, JHP in everything else). Since you asked for a reason, for me it's that ammunition is part of an overall system, and I like consistency to the point of boredom in my systems. My goal is effective hits, quickly. On a human, the JHP will generally be more effective, but if there was no expansion (say due to a HP clogged by clothing) I still get the penetration of FMJ with a bonded bullet. I also use consistency in my training and carry load. I know how it behaves, and how well I can shoot it. I know where the trigger will break and what the recoil will be like. In turn I can get that front sight or point back on the target and go again.
I have a question for those that have answered that they carry both: What kind of barriers will an FMJ in a handgun round penetrate that a JHP handgun round will not? I'm talking the common autoloading calibers here, as the variance between bullet types and loadings in revolvers tend to be a little more stark.