Antimagic field vs 9th-level globe of invulnerability: who wins?

Question

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.)

Antimagic field:

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.

Answer 1

Antimagic Field will prevail

We can look at this from two points of view, both from the rules and from a purely RP perspective. Let's check the rules first.

The description of Antimagic Field states that

Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can't protrude into it. A slot expended to cast a suppressed spell is consumed. While an effect is suppressed, it doesn't function, but the time it spends suppressed counts against its duration.

and that

Spells and magical effects such as dispel magic have no effect on the Sphere. Likewise, the spheres created by different antimagic field spells don't nullify each other

However, the Globe of Invulnerability can be dispelled or suppressed, as it specifies that only what is inside it is protected, and not the globe itself. Therefore it stands to reason that when the two spells meet, Antimagic Field will suppress the Globe of Invulnerability itself, which will allow the field to affect creatures or objects within the globe.

If we look at it from a RP point of view, we can once again turn to Antimagic Field's description, which says of the area of the field:

This area is divorced from the magical energy that suffuses the multiverse.

We could then say that this area lacks the required property to sustain a spell, which would suppress the globe. This isn't a case of a spell being stronger than another, such as light spells vs darkness spells. This is an instance of a spell removing the necessary energy for another spell to sustain itself, thereby making it fail.

Answer 2

Antimagic field wins

Antimagic field contains a final piece of relevant text (PHB, p. 214):

Dispel Magic. Spells and magical effects such as dispel magic have no effect on the sphere. Likewise, the spheres created by different antimagic field spells don’t nullify each other.

This text supports the idea that spells which would suppress or end an antimagic field are not effective on it. There is no similar text giving the same property to globe of invulnerability. In fact, Jeremy Crawford has indicated that globe of invulnerability can be dispelled (although dispel magic could not target anything inside the globe until it had been dispelled).

This provides an exception to the general rule that an 8th-level spell cannot affect something inside a 9th-level globe of invulnerability.