We study of millions of scientific, technological, and artistic innovations and find that the innovation gap faced by women is far from universal. No gap exists for conventional innovations. Rather, the gap is pervasively rooted in innovations that combine ideas in unexpected ways - innovations most critical to scientific breakthroughs. Further, at the USPTO we find that female examiners reject up to 33 percent more unconventional innovations by women inventors than do male examiners, suggesting that gender discrimination weakly explains this innovation gap. Instead, new data indicate that a configuration of institutional practices explains the innovation gap. These practices compromise the expertise women examiners need to accurately assess unconventional innovations and then "over-assign" women examiners to women innovators, undermining women's innovations. These institutional impediments negatively impact innovation rates in science but have the virtue of being more amenable to actionable policy changes than does culturally ingrained gender discrimination.