How to prune this greengage tree?

My parents are visiting this week and my dad has bought a small selection from his power tool collection to tackle some of the bigger jobs in the garden. One of those is the pruning of this greengage tree:



We got about 5kg of fruit from it a few weeks ago so we want to try and do a reasonable job of getting it under control without killing it completely! Any tips on what approach is best to take? As far as the neighbour can remember it's not been pruned in a couple of decades at least - and we're well aware we might be sacrificing next year's crop, but the haircut is badly needed.

 

Prune. Early Spring or Mid Summer.

Unlike pruning fruit such as apples and pears where you prune to flower/fruit buds, plums basically require a general tidy up. With all trees, the center of the tree needs to be kept open. An old practice used to be. That you could throw your hat through the tree.

Judging by the growth. Does your garden run East to West. The idea is to balance the tree up, to cut out any branches that are crossing each other. Get rid of any damaged branches and don't be afraid to cut out some of the older wood. Fresh new growth is to be aimed for.

From your photo, I suggest reducing the top growth and shorten much of the side growth. This will begin to give some basic shape. Then concentrate on those what look like large bundles/masses of branches. Thin these out. Don't panic, the new growth will soon appear. Take your time and enjoy.

 

The tree has a nicely developed trunk and it'll look quite handsome with careful pruning. I'd agree that now is a good time to do it. As Mike says.. new shoots will soon grow next Spring so I'd take a bold approach.
Greengage is underrated, and not over planted, i absolutely love the fruit.