Not at all! The AO+ Mist is not a replacement for soap. We just want to help society understand that there’s a difference between "clean" and "sterile." Our obsession with being as clean as possible has generally resulted in removing all bacteria from our skin, including the beneficial and necessary ecosystem that is our skin microbiome.
When it comes to soap (or your favorite makeup, moisturizer, sunblock, etc. you-name-it), while it's unlikely we've tested the specific product you have in mind, it's okay to use products that are not compatible with the Mist, so long as you remember:
The important things are regular application of the Mist and application at night, when the Ammonia-Oxidizing Bacteria (AOB) have lots of time to operate (they like sweat, and you sweat a lot while sleeping) and when you have little else on your skin (or after anything you've applied has absorbed).
If you're going to shower frequently, using Biome-Friendly products makes it more likely that you will build up AOB on your skin. Mother Dirt Shampoo, Cleanser, and Moisturizer are particularly gentle and allow less-frequent reapplication of the Mist.
You can always try experimenting by using the Mist at night, by itself, for 30 days and then using the Mist in combination with your usual routine for another 30 days.