This calendar is working for the Northern hemisphere and I worked it out with produce in season in this part of the world (Europe parts, not just Ireland).
Not every country will produce all of those but because they are in season, the chance is that they will be cheaper to buy when it is the right time of year.
As well as fruit and vegetables in season this May, here is what you can get all year round: banana, lemon, onion, pak choi and shallots.
Note: this list is non exhaustive, there are quantities of fruit and vegetables varieties available around but I tried to stick to what can be found more often than none.
Also, I included herbs/lettuce and types of nuts in season.