Hi Justin,
It’s impossible to control the way your mobile phone rings when using it abroad on a foreign network.
If you don’t want people calling you to know that you are abroad there are some solutions.
One is called VoIP or Voice over IP. In case you don’t know what it is, VoIP is a telephone service which uses a broadband Internet connection instead of a traditional telephone line to connect.
VoIP services have evolved a great deal over the years and today you can subscribe to really complex, feature rich plans. And one important feature is mobility: you can take the service with you or even use an app on your smart phone. The only condition: having a good connection to the Internet. Of course, when you are not connected you can forward incoming calls to a voicemail message or to a different number (your regular cell number – in this case people calling you might hear the foreign ring.
One of the popular services that comes to mind is Vonage, but there are others too. Check out their plans, talk to them, tell them exactly how/where you plan to use their service and what your concerns are.
Another service you could use is Call forwarding. You can sign up for service with one of the companies offering this type of service (Florida based United World Telecom is a popular one) and they’ll give you a phone number in a country and area code of your choice. When dialed, the number forwards the call to another phone number of your choice, even an international phone number. Usually, these services also allow you to pick the ringback tone people hear when dialing your number.