Hi, seems like the verb tomber most definitely belongs to the set of verbs which can take either auxiliary in the passé composé, depending on their transitive/intransitive usage.
As an example of such a verb, see your very helpful page here:
Monter can be used with avoir or être in Le Passé Composé depending on its meaning in French
Could you please confirm that tomber indeed deserves such a page, and in general remark on whether about 20 other verbs deserve one also (albeit not very commonly used ones?)
Thank you!