Monstera deliciosa seems to fit the bill. From Wikipedia:
Wild seedlings grow towards the darkest area they can find until they find a tree trunk, then start to grow up towards the light, creeping up the tree.[dead reference link]
The fruit of Monstera deliciosa is up to 25 cm long and 3–4 cm diameter, looking like a green ear of maize covered with hexagonal scales.
The fruit may be ripened by cutting it when the first scales begin to lift up and it begins to exude a pungent odor. It is wrapped in a paper bag and set aside until the scales begin popping off. The scales are then brushed off or fall away to reveal the edible flesh underneath. The flesh, which is similar to pineapple in texture, can be cut away from the core and eaten. It has a fruity taste similar to jackfruit and pineapple. The unripe green fruits can irritate the throat and the latex of the leaves and vines can create rashes in the skin, because both contain potassium oxalate: that is the reason why the fruits have to be consumed when the scales lift up.
Plants grown indoors in temperate regions occasionally produce flowers and fruit.
In Mexico, a leaf or root infusion is drunk daily to relieve arthritis.[dead reference link]
In Martinique the root is used to make a remedy for snakebite.[dead reference link]
Here's a picture of the fruit by Bruno Navez:
Dragonfruit (Hylocereus undatus) is also hemiepiphytic. You may also be interested in this page about epiphytic cactus fruits and this post on Yahoo! Groups where I found M. deliciosa. The author claims:
I have eaten the fruits off my Acanthostachys strobilacea, an epiphytic bromeliad, and they are reminiscent of the related terrestrial pineapple just slightly more acidic but still very palatable (to me at least). [Hyperlink added.]