Importing Animations from Blender on Multiple Objects

I have 9 objects with 9 distinct animations in Blender -- separate data for each animation, and they appear as 9 animations in the dope sheet, graph editor, and so on. When I export to FBX and import into Unity, I get 9 x 9 == 81 clips instead of the expected 9. Each clip is repeated 9 times with the name of the object followed by the name of each animation. How can I import with only 9 animations?