Dinarius wrote:
We are based in Dublin, Ireland.

We have a pair of mature lime trees at the end of our garden.

Both trees have a vigorous growth of shoots, about 6-8 feet tall,
around the
base.
[...]

A fence made of pressure treated lumber will outlast you. :-)

Whether you want the volunteers there or not is a matter of taste. Just
do what you want with them. Under Common Law, your neighbour is entitled
to cut back any shoots that grow on his property, but he is not entitled
to use any tree-killing herbicide on them.