It is now clear that The Thing which appeared on my tree, Allophylus psilospermus, is indeed an example of witch’s broom (or witches’ broom in the plural, if you write for the fungus and lichen journal Mycologia).
To refresh your memory,
A Witch’s broom is a disease or deformity in a woody plant, typically a tree, where the natural structure of the plant is changed. A dense mass of shoots grows from a single point, with the resulting structure resembling a broom or a bird’s nest.

First recorded instance in Sapindaceae
The Allophylus tree is in the family Sapindaceae, the soapberry family, and I went looking for other instances of witch’s broom in that family. But Dr. Pedro Acevedo, of the Smithsonian Institution and a specialist in Sapindaceae, wrote to me that this is “…the first time that I know of the occurrence of witches’ brooms in Sapindaceae.”
The agent that causes this witch’s broom is not known, either, unsurprisingly. An expert in fungus, Dr. Meike Piepenbring, wrote to me that she has seen several witches’ brooms on avocado and on native plants here in Panama, but not on Allophylus. She found the presence of, not fungi, but aphids. After studying the literature, she concluded that the most probable agent is for the witch’s brooms that she saw is mites, but she has not yet seen these mites herself.
Ted MacRae, in a comment on my earlier notes on this witch’s broom, had suggested that I might look for mites, and I did. I found one on the first “broom” I examined. I have since, very unscientifically, looked through more than a dozen or so brooms, chopping them up with pruning shears, and have found a few more mites, a couple of tiny beetles, a weird-looking white flat thing that I couldn’t begin to identify, several very small and various spiders, and no fungal spores, at least not any detectable with my 16X hand lens. I don’t have much hope of discovering the causal agent all on my own.
Not so innocent brooms
One little broom all by itself doesn’t look so bad, especially since it has taken the basic shape of the inflorescence and become carried away with it, so that it looks like it might give rise to a robust bloom of small flowers.
But once you know that witch’s broom is a disease and that it could harm the tree (for instance, witch’s broom is a grave threat to cocoa production), they’re not so attractive. Here’s a portion of a tree that is infested with these things.

Maybe even these green brooms don’t look so bad to you, but they look pretty nasty after they’ve “died” or whatever they do after they’ve accomplished their purpose. The brown or black ones may be a little harder to see, even after you click the image on the left. So I’ve circled them in the image on the right, which is also clickable.

No self-respecting inflorescence would end up looking like that!
It is my hope that a graduate student somewhere with an interest in Sapindaceae stumbles across this post and decides to write a thesis on witch’s broom in Allophyllus psilospermus! Wouldn’t that be great?