US infant traveling to Japan: Passport requirements

I am a US citizen and my wife is a Japanese citizen. We live in the US. We have a 7-month-old daughter with dual nationality, and she has a US passport.

Question: Does my daughter need a Japanese passport to enter and leave Japan?

Motivation: We thought this was a no-brainer -- since my daughter is a US citizen, she should be able to enter and leave Japan like any other US citizen, on a US passport. However, we have repeatedly called our embassy (in NY), and they insist that our daughter needs to also obtain a Japanese passport to show at customs in Japan! And they refuse to clarify exactly what would be done with us if we landed in Japan with no passport for our baby.

Update: My impressions from informal conversations with family in Japan, who spent considerable time asking questions at the Shiyakusho ("city hall") lead me to tentatively conclude the following:

(a.) If we go to Japan with the baby and no Japanese passport, they will let us through customs with no explicit penalty.

(b.) There is a risk that our behavior will be interpreted as an implicit rejection of our daughter's Japanese citizenship, making a barrier should she ever wish to be treated as a Japanese citizen for any reason.

Again, these are only second-hand impressions. Continued discussion is appreciated.