So the width of this strip is so narrow that you can't actually walk down it to the bottom, to be able to erect a new fence from your side?
If so, what's the point of 'owning' this land - you can't do anything with it.
If your land is registered at LR and the little red line bounding your plot on the title plan does not coincide with the line of the garage, then there can be no doubt that you do indeed own some land beyond the outer wall of the garage. However the LR plans are called 'general boundaries' and the LR are very clear to state that they are not exact boundaries and one cannot scale from the title plan. You will never 'lose' the land (whatever width the strip is) because with registered land, any attempt by the neighbour to make a change to the title plan will result in the LR writing to you to ask if you agree with the new boundary.
However, more than that, you would be well-advised to avoid falling out with the neighbour over this - such issues that have ended up in courts have cost the parties tens of thoudsands of the pounds. Often all for a tiny strip of land.
So the width of this strip is so narrow that you can't actually walk down it to the bottom, and hence use it for anything?