In Illustrator I can't align an object with a stroke with an object without a stroke. It keeps popping over out of alignment. How can I stop it from popping over?
This is most likely because the non stroked item and the stroked item are slightly differently placed in the pixel grid.
One of them is probably going to half pixels. The other is not. When you try to make them align, one will always push out to a pixel position that suits itself, which will be different than the other, and they simply can never truly lineup.
Try reducing the size of the shape without a stroke by just one pixel in both axis. This might force it to align with the edge of pixels instead of their midpoint.