I was born on the other SBA, at Dhekalia, and I'm afraid she'll just have to grin and bear it. Although yes, she was born on 'British' Soil, that happens to be on Cyprus.
There isn't anything she can do about it, it can be a problem on occations, insurance companys for example have difficulty understanding it, but you get used to it.
On thinking about it, there is one thing that helps, I usually put in large letters 'BRITISH MILITARY HOSPITAL' Dhekalia as place of birth. So try RAF Hospital, as TPMH wont make sense to most poeple.