The Expedition desk in the Anomaly is pretty slick, but if you collect stuff with wild abandon during an expedition (don't we all?) you will have some interesting questions to answer at the end!
The mechanism is: You claim all the rewards, including the new ship for completing all 5 phases, which, you have to be outside of your ship to claim, so proooooooobably you will do this in the Anomaly. Then head for the Expedition desk, where you can click and confirm that you want to end the expedition (it will warn you that you have incomplete optional tasks). That will open up the "transfer back to main save" storage. Which is interesting because: it isn't any larger than when you started. So if you started Omega from your main save and brought a bunch of stuff over, and you didn't use that stuff (I brought 40 repair kits, for example), then you have to decide if restoring those items to your main save is more important than bringing back any new stuff you collected (memories. hyaline brains, anomaly detectors, storage augmentation, lots of expedition related items with way more value than another 5000 carbon).
Once you've made all of those decisions, you again click to "finalize expedition". At this point, it will convert anything you aren't transferring back into units, nanites, or quicksilver! There is a nice display that lists a bunch of statistics, see screenshot (assuming I've attached it correctly). This is a good reason to max out your expedition suit and ship storage and Do all The Things.
One curiosity: Since I've got thousands of hours in this game, I am always running close to MAXINT units in the bank, so the calculation of how many units I earned bailed out when it hit the threshold, and just says -1. HOPEFULLY when I click this here button, I will still have my billions...if it actually resets my units to negative 4B...uh that's a bug.
On the other hand: 20k nanites and 9k quicksilver?! Hell yes!
Verdict: It's not perfect but I really like the new expedition desk and the whole "bring your ship and MT and a handful of stuff with you" capability.
Very cool! I thought the ship rewards were special and didn't know it applies broadly. Thanks.
As with all past expeditions, all unique expedition items (ship, staff, posters, etc) can be redeemed at the quicksilver synthesis station in any save. It's just the resource rewards (i.e. things you can stuff in your inventory) that you have to pass back through the new store-and-retrieve mechanism.