A beautiful example of what this might mean is the C# minor Exercise by Clementi in Preludes and Exercises - a truly magnificent source of lyrical insight into the piano, the hand, tonality, you name it.
The exercise's meter is 3/4, the tempo Allegro ma non troppo. I maintain that this means that the quarter note is not the structural metric unit, except perhaps when it is. There are seven, count them, seven quarter notes in the whole piece. The measures are either clearly divided into halves (dotted quarter plus six sixteenths) or some marvelous unlikely combination of asymmetrical rhythms like 6 eights in one hand plus ten sixteenths after an eight rest in the other.
I propose playing such an unlikely measure as two groups of three eighths in one hand, with two off-beat quintuples in the other.
This was for a student whose rhythmic command is far from reliable. He played better the more complex the task. More important still, he had a blast doing this impossible coordination.
I had him play in strictly emphasized quarters just to feel the difference.
Such a piece convinces me that Allegro refers to a metric proposition wherein the quarter note has distinct structural function. A modifier like ma non troppo indicates that the quarter note is theoretical only, not actual.
