We have been struggling to identify this tree in our garden (in Devon). Through every aspect it would appear to be of the lime family, however the leaves are neither cerated nor fury on the underside. When it flowers they are white and the berrys are of a dark colour. It appears to have been chopped back at some point but currently stands at around 20f with a very large span. Any help identifying this would be greatly appreciated. It happens to be shading a large portion of the garden currently and we don't know how it should be dealt with.
Personally, though in the absence of a lot of information about your garden, I would initially see (picture mentally) if the problem could be solved by 'raising the crown'. That means pruning off the lower branches but leaving the higher ones. That would let light in sideways - which might be enough to let other plants grow there better, though you would need to select ones that tolerated dryer conditions. (But the area could be a stream bed for all I know, so dryness might not be a problem.)
Alternatively, heavy pruning would let the light in temporarily, but you'd lose the flowers and the character of the tree, and it would be a job to be repeated every few years.
If it has to go, consider retaining the trunk and some of the branch stubs, with a view to growing a climber up it. You would have to kill the tree chemically.