@JasonStarr Can you point me toward a reference? Also, it's not clear to me how this would get one closer to an elliptic surface, but maybe I'm just not seeing it. The only way I can think of is to start with an elliptic surface without a section and with multiple fibers, then do a log transformation. But then of course, determining what one ends up with is nontrivial... Curious to see if anyone can point me toward a specific example of this.

I do need it to be projective. Edited accordingly.