How to see compass accuracy and recalibrate the compass in iOS 10?
In troubleshooting this, I learned a few important things:


While there is no way to manually force a recalibration of the compass aside from toggling location services on and off, generally the compass attempts to recalibrate itself when an app requests it if it has not been in use.
The 'blue cone' in maps seems to be the best way to easily tell how accurate the calibration things it is.
Arguably most importantly (and most embarrassingly), my issues were entirely caused by a case that I was using that had a small magnetic clasp in it.  Everything worked just great after getting rid of it.