Yucca Gloriosa side shoots

This sounds like it is multiplying by producing small yuccas. They do this. You can divide if they are too big (thorny business). They can be pretty tough, and if you try to move them the roots are deep and more yuccas come up from the old roots. I speak from experience of inheriting a front garden full of them. I do live in an area where they seem to do very well, mind you.

just cut the dead flowers off, you can even chop off the rest of the shoots and use as cuttings