Greetings, welcome to the Forums.
Your 'Coral Cactus' is some type of grafted succulent, combinging a difficult-to-root crested scion with a vigorous green rootstock.
Without seeing the plant, it can't be identified with certainty, but many common specimens of 'Coral Cactus' are not true Cacti at all,
but rather a graft of two succulent Euphorbia. Most commonly seen is a crest of Euphorbia lactea 'Cristata' grafted onto a stem of of Oleander Spurge (Euphorbia neriifolia).
In your case, as is often happens with grafts, the rootstock has begun to put out branches. You have several options.
Some succulent growers consider it prudent to always remove rootstock branches lest they overwhelm the scion. However, I have seen both Euphorbia and Cactus grafts where the crested scion has persisted and grown well, either despite or perhaps even because of the vigorous green, branching rootstock it is attached to.
If you decide to keep the rootstock branches you must either transplant or set the plant into a larger, heavy pot that will keep the plant from tipping over.
If you decide to remove the rootstock branches, cut them off with a sharp knife. Make sure you wear rubber gloves and goggles when performing this operation, as the milky latex of Euphorbia is both corrosive and toxic. Of course, you can then subsequently root the severed branches to make new plants.
