who wants to diagnose my car?

its bung.

its a 1986 corolla, and runs fine-ish, except for the following problem.

if ive been driving it for anything longer then maybe 10mins and i stop for a short time, as in to get petrol or something, it doesnt like to start again.
i turn the key, foot all the way down, and it will catch, and just barely tick over, but wont really fire. if i take my foot of at all it stalls.
It seems fine again after leaving it for 15 minutes or so.

Any ideas?

Possibly fuel vapour-lock? Injection system getting too much heat form exhaust and boiling the fuel in the fuel rail or carb?

Fuel pump?

could be a number of things but sounds like a fuel issue, also check ur fuel filter, might be clogged if its old and get them to check the AFM, might need a clean, wont hurt anyway