Odile by Sibylle Hagmann.

Related:
http://typophile.com/node/4088
http://typophile.com/node/9487

Not available commercially, but Emilie Rigaud's Coline is made of three faces with varying levels of cursivity.

My graduation project this past year was an experiment with integrating multiple italics. During research for this project I was inspired by Fairfield which had two italics drawn for it, with two types of cursive structure.

(more information about Arietta here)

(You should also link to what seems to be a wonderful Arietta specimen book.)

Warren Chappell’s Lydian has an Italic and a Cursive.
The styles of Kursivschrift have different angles, but all of them are cursive.

Very useful thread - thank you!
There's also Seria, but I don't think its second Italic is released yet.

BTW, although Fairfield and Electra (and in a way, Centaur) featured
two italics, the first font that did so unapologetically was Treacy's Arrow.

hhp

Danubia has a normal and a script italic.

I’m really looking forward to the new Seria Italic. It’s such a beautiful face!

Most of all, Seria needs a real Italic instead of that upright affectation.
"I just wanted to make one" indeed - as if users [should] care.

hhp

Perhaps the brand-new FF Masala by Xavier Dupré falls into this category ...

The original ATF Cloister has two italic versions, one simply called Italic and the other, Cursive. The latter featured swash caps.

And if Masala fits here, so does Fresco, with its italic, informal, and script variants.

Then Fertigo too.

I can't think of any actual examples offhand of types supplied this way, but this could be the result of the fact that typefaces can be distorted both in typesetting machines and on computers: I had thought that one way in which it is actually fairly common for a typeface to have two italic versions is where in addition to a conventional italic, with some script-like characteristics, one also has the option of using an italic which is just the original Roman font, slanted.

Also, with Venetian Old Style fonts, like Bembo, the companion italic is usually a separate font the letters of which are much narrower. Some such fonts might have a conventional italic as an alternative more palatable to modern readers, but again, I can't cite an example.

Speaking of Bembo, there were two Monotype italics for it, in metal.
AFAIK the Narrow Italic designed by calligrapher Alfred Fairbank in 1928, the one with the almost upright stance, has not been digitized.

Does anyone have additional leads on type families with more than one italic? I’m specifically interested in typefaces designed for book settings along the lines of Seria, which doesn’t appear to be commercially available yet.

Matt, there are two additional possibilities I know of from students in the master's programs at the University of Reading and KABK: Joke Gosse's Melville and Emma Laiho's Taiga. Not sure about the licensing situation with them, but you could try getting in touch.

Since I was just looking for multiple-italic faces, I'll add two more to the thread:
Laure Afchain's Malaussene (student typeface at KABK; translation version released at Gestalten) and Florian Schick's Mag Grotesque (student typeface at KABK).