Alice the wizard casts globe of invulnerability at 9th level. Eve the enemy wizard casts antimagic field and walks to within 20 feet of Alice, causing the 10-foot spheres of the two spells to overlap.
What happens in the overlapping area? Which of the two spells, if any, is in effect in the overlapping area? Does anything different happen if Eve walks within 10 feet of Alice, bringing her inside the barrier of Alice's globe of invulnerability?
The relevant text from the two spells is:
Globe of invulnerability:
the area within the barrier is excluded from the areas affected by such spells
(Note: because the globe of invulnerability spell was cast at 9th level, in this case "such spells" refers to any spell of 8th level or lower cast from outside the barrier.)
The area of another spell or magical effect, such as fireball, can't extend into the sphere. If the sphere overlaps an area of magic, the part of the area that is covered by the sphere is suppressed.