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First create a normal view controller in your storyboard. Then in a button click method, or whatever, initialise it:

let storyboard : UIStoryboard = UIStoryboard(name: "THE_NAME_OF_YOUR_STORYBOARD", bundle: nil)
let vc = storyboard.instantiateViewController(withIdentifier: "THE_IDENTIFIER_OF_YOUR_VIEWCONTROLLER")

Next set the modalPresentationStyle of it to popup, get a reference to the popup presentation controller, make this class its delegate and say it may show only an up arrow.

vc.modalPresentationStyle = .popover
let popover = vc.popoverPresentationController!
popover.delegate = self
popover.permittedArrowDirections = .up

Next set the source view and rect to whatever initialise the popup. In my case it's a sender object in a touch listener.

popover.sourceView = sender as? UIView
popover.sourceRect = sender.bounds

Finallly present it with present(vc, animated: true, completion:nil). It will take up the size of a new screen, unless we're on an iPad, regardless of our source rect and view.

To ensure we only take up the space under the aforementioned button, add this delegate method:

func adaptivePresentationStyle(for controller: UIPresentationController, traitCollection: UITraitCollection) -> UIModalPresentationStyle {
    return .none
}

It may not take up all the space of a new screen, but it will still take up everything below the button.

You can size the view controller with use preferred explicit size checked or do something like vc.preferredContentSize = CGSize(width: 200, height: 100), but I'd like a better method...