We don’t have sharing consents across different top level domains. However, you can use our APIs to get the information about consent and to apply it automatically. This would require using some custom code to pass the consent string via URL param.
Could it be possible to use two different sites to have the consent not requested on the second domain (a domain for our app where you are anyway identified) but to still be able to use cross-domain tracking across the two sites using the pk_vid parameter?
No. I was thinking about using the same property and container code for both domains. One goes by default with the consent popup enabled and on the other one you accept it in the background via our APIs. In this case, cross domain tracking will be working fine.