Hello been a while since i've posted here but had a question, an issue came up in an edh game regarding the interaction betweenObeka, Brute Chronologist and Treasure Nabber. If Obeka's ability is used to end the turn on the turn when nabber's effect ends does obeka's controller keep control of the artifact, or does it go back to the original controller? is there anyway for obeka's controller controller to keep control of the artifact that was taken due to nabber's effect?
No, there is no way to keep control of the artifact after "your next turn" using just Treasure Nabber and Obeka.
If you gain control of an artifact with Treasure Nabber, then end your next turn with Obeka, this will move the turn to the cleanup step (C.R. 717.1d; see also C.R. 717.2e), and the effect of Treasure Nabber's ability will end during that cleanup step (see also C.R. 514.2). It doesn't matter when your turn ends this way; even if you somehow manage to end the turn with Obeka during a cleanup step, this will only move the turn to another cleanup step and the effect will end during that new step (C.R. 717.1d; see also C.R. 514.2, 514.3a). See also this thread and this thread. Compare this with an ability that says to do something "at the beginning of the next end step" or "at end of combat"; see also this thread and this thread.
If you gain control of an artifact with Treasure Nabber, then end your next turn with Obeka, this will move the turn to the cleanup step (C.R. 717.1d; see also C.R. 717.2e), and the effect of Treasure Nabber's ability will end during that cleanup step (see also C.R. 514.2). It doesn't matter when your turn ends this way; even if you somehow manage to end the turn with Obeka during a cleanup step, this will only move the turn to another cleanup step and the effect will end during that new step (C.R. 717.1d; see also C.R. 514.2, 514.3a). See also this thread and this thread. Compare this with an ability that says to do something "at the beginning of the next end step" or "at end of combat"; see also this thread and this thread.
EDIT (Dec. 15; Feb. 25): Correctness edit.