Down side on buying crosses like mine, even if you did find one with the exact same parentage, there's no guarantee it will be exactly the same as mine. That may be why it doesn't have a registered name yet - they have to cross them enough times to dependably get the same characteristics before the Orchid Societies will register its own name. Using a parent like Edisto with a lot of different sports (as Jim said above) you could get a lot of variation for a long time - years and years possibly, as each time they do a cross they have to raise a plant - or batch of plants - from seed for 3 to 5 years before they bloom to see what they've got. Try this if you're curious. Google "Blc. Edisto" and take a look at the various sports of just that one parent.
That may be why it doesn't have a registered name yet - they have to cross them enough times to dependably get the same characteristics before the Orchid Societies will register its own name.