First detach the mushroom from the rock at the base of the stalk.
The slice the mushroom across the radius like a pizza making sure to try and get some of the mouth in every slice. Ricordeas can be fragged the same way as mushrooms.
I have a ricordia that has 2 mouths about 1/8" apart. How well would it work to just leave it on the rock and cut it in half right between the mouths? I dont want to kill it though....
If it has two mouths then it has already started the process of self-propagation. Soon it will become two mushrooms. If you still want to propagate the coral by cutting then just cut between the two mouths and each half will become an individual mushroom.