Actually, that is what's called a bed turret. A tailstock turret looks like a tailstock with one of those turrets on an MT arbor permanently installed, only beefier.
Back when Atlas used to sell tailstock and bed turrets, by default they came in with the tool holder holes all drilled undersize, usually for 3/4" tooling. And with instructions of how to mount a boring bar in the spindle to finish cut them both to size and in proper alignment with the spindle. If the turret assembly originally came with the lathe, that would have been done at the factory. As yours is apparently off front to back, the turret is probably not original to that machine. However, whether the front to back offset is adjustable or not, I do not know. On the tailstock turrets, it generally is by the same method as it is on a normal tailstock. But I don't know about bed turrets as I've never had one.
However, once you get done with anything else that you plan to do to the lathe, your first step with the turret should be to confirm that the indexer and the fore and aft movement lever or handwheel works properly and is properly lubricated. Next step is to check the alignment. If you have a centering type indicator, that works best. Otherwise you will need a mirror to read the indicator through part of the rotation.
Lock the turret to the bed. Set the feeler for the hole, run it in 1/8" to 1/4" and check how far and in which direction the center of the hole is off. Then run the indicator feeler in to near the bottom of the hole and repeat. Note where the high spot is in each position.
If the error is the same front and back, the indexer is probably working OK but the turret is not matched to your lathe. If the front to back position is adjustable, adjust it.
If the error is different front to back, the indexer is off. If it is adjustable, adjust it. Then repeat the test to determine the direction of any front to back error.