Boy, that's a tough question. I don't know of a tomato that performs well in part shade. That said, the situation you describe should produce enough tomatoes providing the variety you choose is a good producer to begin with. There are other factors which may impact production as much as sunlight. They include zone, size of the pot, tomato variety, fertilizer, etc. Also, how do you plan to use the tomatoes? Do you need enough for canning, just enough for fresh eating, enough to sell? Don't mean to be difficult. Just trying to be helpful.

I grow heirlooms mostly and I use 12 - 16 gallon pots. Tomatoes may be grown in 5 gallon pots, but with limited sunlight, I certainly wouldn't try it. The production would be very low. Here are a few dependable varieties that have excellent flavor and decent output.

Cherry - gardeners delight, black cherry, galinas, sungold
Paste - opalka, san marzano
Beefsteak - goliath, Marianna's Peace, Ponderosa, Mexico, cherokee purple, Pruden's Purple ...gosh there are sooo many.
Small - Eva's Purple Ball(my favorite small tomatoe for flavor and production), bloody butcher, break o' day

Also, you may want to check out the tomato forum. Your question has probably been asked there.

Thanks so much for putting me onto the tomato forum. I had no idea it was there.
I'm pretty new to this site, so I'm still getting to know it.
Thanks for your info as well. I would like to try some of the beefsteak variety's, especially the 'unusually' colored ones. Ive been told they have a great flavor.
As for the zone I'm in... Im in Barbados, which is in the tropics. Apparently its a zone 11, Ive never heard of it dropping below 20C.
Thanks again
Ang