I have found the cause and solution to this issue.
The cause is that SystemUI has lost its permission to draw on top of other apps, either due to a manual application permission reset (in my case) or because of an update or rogue app which reset permissions. Here's how to fix it:
Open Settings
Open Application Manager
Tap the three white dots in the top right
Select "Apps that can appear on top"
Tap the three white dots in the top right again
Select "Show system apps"
Scroll down to SystemUI and change it from "Off" to "On"
In my particular case, I also had an issue where if I pressed the home button during a phone call (in order to retrieve a phone number from my contacts to give to my caller, for example), I would get the message that "Phone has stopped." The same solution worked to fix that, as well.
Please let me know if this fixes the issue.