I'm not sure about specific hosts, although if I remember correctly Hostgator offers servers on different continents and mirrors on at least two, but off the top of my head those details may be off.
Another option though that you might consider is a CDN(content distribution network), which mirrors your static files in different locations globally, allowing for shorter response times. I've been testing Cloudflare(https://www.cloudflare.com/) which offers free and paid plans, the free plan is pretty extensive and the difference it made on the site I was testing it with was pretty significant.
Added bonus aside from speed is that it helps add a layer of protection to your server since the nameservers will be Cloudflares and much of the traffic requests won't actually go to your server, that also aids in cutting down on bandwidth.
Aside from Cloudflare there are countless free and paid CDNs that would help with your situation, even with a host that mirrors globally.