Basically, it’s a feature of the VSTi instrument … or not!
A few instruments, for examples Groove Agent 2, Bonnemark (sp?) Bass, Jamstix, etc., can also stream the midi they are using to trigger their internal samples, thus permitting the midi to be recorded. This is because they already have midi generating engines inside them that are themselved activated by midi note messages. They have been specifically designed to pass this data along.
While midi streaming makes a lot of sense in a drum VSTi or a Bass line VSTi … it doesn’t really make a lot of sense in a synth.
I’m not aware of any VSTi synths, including Omnisphere, that do this … unless you want to count the dinky ones like in The System that are primarily intended to help compose midi.
I do believe Omnisphere permits you to export as midi files anything you create in the arppegiator … but this is an entirely different thing. You can take those midi files and then use them as a template/ framework to recreate your own equivalent using ArpacheSX of Arpache5.
Just place that midi arp file in Cubase, duplicate it for a couple of bars, cycle play it with an instrument you like.
Now set up the same instrument feeding another midi track, insert your midi arp plugin in the inspector and just keep manipulating until the one you are working on matches the one you took out of Omnisphere. Save it as a preset for ArpacheSX or Arpache5 (or any other midi arp plugin you might have), and ‘presto!’ Now you have the same arp voice for anytime and any instrument you care to use it on.