Thanks for the answer, @cmcd!
Here’s pp_check() with trait.count ~ (1|species) (without the covariance matrix), as compared to trait.count ~ (1|gr(species, cov = A)):
Looks like the covariance matrix indeed adds a lot of variance, as you suspected. What you’re suggesting is that for some reason the phylogeny might not be fully capturing the correlation among data, thus leaving part of the variance unaccounted for?
(By the way, I don’t even know if this is an issue at all! I just thought I’d ask for help because the overdispersion thing was bugging me, but maybe this is expected in Poisson GLMMs with covariance matrix?)