I figure that on a bike on tour you really only need a pretty general idea of direction since you are following roads. Something +/- 30 degrees is probably sufficient most of the time. A compass can be pretty bad and still provide that level of accuracy as can a general idea of direction based on time of day and direction the light is coming from even on a cloudy day. When there are other factors like visible mountain ranges or other clues just keeping track of which way is north becomes even easier.

Bottom line... Unlike when in the back country or offshore in my sail boat, on the road a compass is usually unnecessary. The few times it might be needed it is a very general idea of direction is good enough. So no worries unless the compass is very bad.