Maybe it depends on where you're from? In my Greek head-accent "e" and "a" are very different. Same with "their" and "there" (the-ir, the-r).

When I speak, of course, they sound identical, but my spoken accent is not the accent I think in.

I don't think it's a question of sound. (I'm french) 'they're', 'their' and 'there' sound the same (spoken or in my head), but their meaning has nothing in common so I never mistake one for another. How you acquire a language plays here. I learned english fully conscious of what I was doing rather than my organic absorption of french.

Oh, of course. I'm just saying that the most common reason for conflation (the fact that they're homophones) doesn't exist for me.