Localization dependent on prefix and front page dependent on domain

We are using the module internationalization to translate our content.

I wonder if following is possible to do with internationalization:

Our web site is available in two languages: German and English. German is default language and I'm using the preferences "Path prefix with language fallback" to decide which language should be taken.

Our web site is also available by using different domain names: An English one and a German one. Now I'd like the visitor to see the English front page if he uses the English domain and the German front page if he uses the German domain. But normally he should be able to switch between languanges and the language should be dependent on path prefix with language fallback.

I hope somebody can help. Thanks