The emulsifier might make the soap look opaque (creamy?) but it will kill the lather.
If you want an opaque soap, then make the soap with fats rich in stearic and palmitic acids such as the shea that Ali suggests. And/or superfat later on with stearic acid as
@IrishLass does with her Coco-Shea soap.
If you want to thicken the soap whether it's opaque or not, you can get a decently thick soap just by diluting the paste with care. I think xanthan gum can work too, but people have problems with it. There are other types of thickeners that are easier to use and more reliable.
I can't say I find any lye-based soap to be "conditioning" in the sense that it soothes, lubricates, and protects the skin. That's more the realm of a body lotion or hair conditioner.
But it is true that some soap recipes will be milder to the skin than others, which might lend the perception of the soap being more "conditioning." Since we don't know your recipe, it's hard to give advice about that.
Superfatting with a fatty acid (stearic acid for example) is one way to add mildness. Making a "combo cleanser" which is a blend of lye-based soap and synthetic detergent can add mildness. If you choose the right syndet, that detergent can also increase thickness.