Produce in Melanesia

Fiji - most small villages we visited did not have organized outdoor markets; there seemed to be little formal trading between families or villages since everyone was growing the exact same thing.  Once or twice we received gifts or trades of homegrown produce from a friendly local.  In larger villages and towns there was often a Chinese or Indian shop with some basic processed foods and freshies. 
When we found a market we could usually choose from cabbage, lettuce, tomatoes, eggplant, onions, papayas, bananas, plantains, limes, breadfruit, coconuts, taro, cassava, and kava.  The best-stocked markets also offered watermelons, pineapple, capsicums, squash, beans, fern fronds, raw peanuts, and spring onions.  The local diet often is limited to unseasoned fish and taro and seems bland to the extreme.

Vanuatu markets were a little easier to find, even in the small villages.  If there was no formal market there was usually someone who was interested in selling produce from their garden to passing yachts.  Choices were roughly the same as in Fiji, though as we moved north the variety improved and goodies such as ngali nuts, yams, sweet potatoes, capsicums, star fruit, oranges, island pears, and island cabbage became easy to find.  Vanuatu prices were a bit cheaper than Fiji ones.

Solomons - Honiara has one of the biggest produce markets in the area.  Here we encountered Kumara (sweet potato), cassava, taro, yams, manioc, pana, ngali nuts, cutnuts, betel nuts, bush limes, mandarins, oranges, lemons, pommelo (oversize grapefruit), breadfruit, star fruit, mangoes, opiti, pineapple, watermelon, pumpkin, snake beans, wing beans, runner beans, pak choi, saladeria (both types of Chinese cabbage), slippery cabbage, swamp cabbage, English cabbage, fern shoots, pumpkin shoots, spring onions, eschallots, sugar cane, pawpaws, bananas, plantains, coconuts, avocados, and more…
Outer islands often featured markets or trading canoes offering betel nut, bread, bananas, papayas, pineapple, peppers, tomatoes, squash, and various greens such as western cabbage, island cabbage of the dark green and light green varieties, fern fronds, and a horseradish-like green vegetable.
Solomons prices and variety have been the better than Fiji and Vanuatu so far.  In particular, it seems that one can always get island cabbage, pineapple, and watermelon.  We have not seen any kava, Island pears, lettuce, or western cabbage here yet.