i opened an offshore account with Lloyd's from NZ but I would advise you against doing that. it costs $100 + in processing fees, takes 4-6 weeks to be processed and once you finally get it you find out that the offshore account has much higher fees than a standard/classic account. i ended up closing my offshore account and opening a classic account within a couple of weeks of getting here, and the hassle of opening a classic account was not made any easier by me having an offshore account with the bank already, in fact it made no difference whatsoever.
i also suggest you avoid Lloyd's altogether. HSBC is much better and much more efficient I'm told.