This is a word I ask for in my dialect surveys in rural China.

The name for caul fat in many Chinese dialects is "net-fat" (Mandarin wang3-you2), because of its reticulated appearance. (The fat itself forms what look like a network of white filaments within the membrane that bears it.) I think this term is more likely to persist in Chinese than the English equivalent "caul fat" is in English, because few people today have ever seen any sort of caul. The vividness of a term depends on its referents.

In the Chinese countryside, as in most traditional societies, every part of the slaughtered animal is put to some use. The caul fat of the pig is used medicinally - I am told it is fed to very young and very old people suffering from weak constitutions. Both the fat itself and the surrounding membrane are considered to be medicinally efficacious.

Weaving my way through six degrees of node separation from this node after posting it, I came upon the two words epiploon and omentum for this thing. Epiploon I have not seen before, and have nothing to say about it except that it is four syllables long - rhymes with go on rather than buffoon. But omentum I have seen before. I have seen strange forms of meat-food packaged fresh in Vietnamese supermarkets in St. Paul, Minnesota, labelled omentum. I stuck the word away in my mind and there it slept. So now I realize it is the same thing I've seen being taken from newly slaughtered pigs in the Longyan countryside.

Caul itself was apparently also used for this peritoneal membrane. What a joy it is having Webster 1913 on tap right here!