That's an interesting question. I'm not a teacher or grammarian, just a U.S. native speaker. I see your point. The sentence sounds natural to me, and I would accept it as correct as written, even in formal writing. You could rewrite it in either of these two ways:
"However intricate ARE the ways in which animals communicate with each other, they do not indulge in anything that deserves the name of conversation."
"However intricate the ways in which animals communicate with each other ARE, they do not indulge in anything that deserves the name of conversation."
Yet it sounds or feels stylistically better to me without the word "are." I don't know how a grammarian would parse or explain the grammar. Perhaps this is a case where we abbreviate a sentence by leaving out a word that grammatically is necessary, because we can understand the sentence without that word.
If I were going to add something to fill out the grammatical structure, I am not sure whether I'd choose "are" or "may be."
The following sound and feel correct to me as written, too, although one could insert "is" or "may be" just before the comma:
"However heavy the rain, eventually the sun comes out."
"...there is no circumstance noticed which shows, however strong the mutual sympathy, the slightest influence of particular attraction. " --1830's English from a biography of the poet Byron
"However bad the situation, there is always something we can do."--from someone's blog post.