mmccomb's solution eventually worked for me, but I had to tweak a couple of things. Hopefully these details help someone:
1) I had to set the property of the presented view controller from the presenting view controller, like this:
MyViewController *myVc = [segue destinationViewController];
[myVc setModalPresentationStyle:UIModalPresentationOverCurrentContext];
Setting the presentation style from within the presented VC's viewDidLoad didn't work.
2) I was using a storyboard, and I had to go into the segue properties and set the Presentation style to Default. Setting any other value gave me a black background.