I finally (fig)ured it out.

There's a fruit tree near the entrance to my house and from time-to-time, my host father picks some of the fruit and brings it inside. I usually have one and I figured it was some sort of exotic fruit native to Japan. Today I asked my host parents what they are called. It turns out they're just figs, or ichijiku (いちじく). I guess I'd never seen or eaten a fig in the US. They taste somewhat like fig newtons, which isn't surprising. You learn about all sorts of things studying abroad.