When does one cut off the large flowerless portion of an Orchid?

We have an orchid whose flowers have come and gone. The plant itself is still very healthy, there are roots shooting out all over the base of it, and the leaves are still bright and green some 4 months after the flowers have fallen off.

So should we cut the large green stem now? Will another one grow? Orchids are intriguing flowers, but a complete mystery to me as to how to keep them properly…

Sounds like a Phalaenopsis, however one spells it.
Cut the bloom spike above one of the scaly looking places that a bloom grew from; with any luck, it will produce a short bloom spike to produce a few more blooms.