This quiz seems to have ran its course, so here's what I know.
The species is Chrysophyllum gonocarpum; common name is aguaí (PT_BR), aguay (SP).
It is found in the subtropical parts of south S. America.
The tree is rather nice looking, with its long deep green leaves.
It does take a bit of cold and does not show any kind of stress during our Mediterranean type winter.
It does not like alkaline soil and I need to give it chelated micros from time to time.
The fruit is rather small, about the size of a large grape. It is rather seedy. The seeds
are very telling of the genus/family.
The pulp is sweet. Unlike Helton Josué's (from Brazil) description, the pulp is not dry. It is moist and tasty. However, there's something in the skin/pulp that is irritating. I ate a bunch of them and soon afterwards I started feeling a burning sensation in my mouth. Fortunately, it did not made me ill in any other way. Pity the fruits are small and cause that strange sensation.