Importing Illustrator leaves gaps between objects

I made a file in Illustrator, a file that has many paths. In Illustrator, these paths overlap and do not leave any gaps. When importing it to PS, I see gaps that are maybe 1 or 2 px wide, that I then have to fix manually. The import should not have these gaps. I checked, and I am seeing the PS file at 100% magnification.

Interestingly, if I add a background to the PS file of any color, the gaps disappear. But when I export it to PNG, without the background, the gaps persist.

(Note: I am trying to attach the Illustrator file, but I get an error saying: "The attachment does not match its file extension and it has been removed")