Continuity mistake: The Supreme Intelligence has a worried expression when Carol gets close, and drops the energy beam that was not working. However, inexplicably she's smiling once Carol rips the control device from her neck, before switching back to the kind of "Oh no" face she had before. (01:31:45)
Continuity mistake: When Carol asks "Arm wrestle for the tesseract?" Minn-erva in the background is holding her gun at a different angle than the previous shot. Following that, Bron-Char (the big white bearded Kree) assumes battle stance at least 3 times in different shots, at first reacting to Carol flipping on her feet (but looking composed in the next shot), then shown with his fist stretched out, then shown in the process of stretching said fist out. (01:33:33)
Continuity mistake: Talos has been hit and Fury asks him if he's okay. He answers "Never...better", hand on his own chest. In the next shot Fury is keeping the hand on his shoulder and Talos has his hand on Fury's arm. (01:38:30)
Continuity mistake: Dining with the Skrulls at the end of the movie, Carol responds to young Monica's "You don't know" with "He doesn't." Sam Jackson is holding his finger on the glass differently between shots. (01:47:49)
Continuity mistake: When Carol is showing Fury the upgraded communicator, in the close-up Fury is already drying his hand with the towel he grabs only later. (01:48:40)
Continuity mistake: Going through the old file about Carol's case at the end of the movie, Fury reaches for her picture, which is turned 90° to fit in the folder (being horizontal) but at the cut the picture is already oriented correctly for him to look at. (01:52:20)
Answer: Howard Stark relinquished custody of it to her. For a full history of the Tesseract see here: https://marvelcinematicuniverse.fandom.com/wiki/Tesseract.
ctown28 ★