In Gentry's easy FHE intro, it is stated that
Researchers [1, 8] showed that if $\epsilon$ is a deterministic fully homomorphic encryption scheme (or, more broadly, one for which it is easy to tell whether two ciphertexts encrypt the same thing), then $\epsilon$ can be broken in sub-exponential time.
Side question: This answer mentions that any probabilistic PHE scheme can be made deterministic. This holds also for FHE schemes right? Any implementations out there make this easy to do?
Main question: Are there any FHE schemes that meet the "more broadly..." part of the quote? Specifically, are there any FHE schemes out there that allow for easy determination of whether two ciphertexts encrypt the same thing?
