The Vatican does not really have a need to cooperate with other churches, similar or not. I know that there's a lot of interfaith communication, but these various Christian ideologies are entities unto themselves. That's the whole point of having different sects within a belief system. It's not up to the Roman Catholics to make you feel better about choosing Eastern Orthodox.
I do not believe it matters if the Vatican cooperates with other similar churches such as the Eastern Orthodoxy. Although a lay person may see the similarities, it doesn't mean they are similar or that they should work together. I think assuming churches will work together is wrong, given that all of these different denominations and sects exist due to disagreements.
Overall I think the Vatican has a cooperate attitude towards other sects of the religion and other religions in general the Vatician's main job is to project an image of cooperation and peace in general so treating other similar churches with respect and cooperation is something I think is key for them to do.
The Vatican does not cooperate enough with other similar churches such as the Eastern Orthodoxy. They do not work well when talking to each other about certain situations. I feel that a lot is hidden between the two churches that they don't want the other to know about making it hard to communicate right and wrong.
I do not think that the Vatican cooperates enough with other similar churches. I think that the Vatican does not do enough to start a dialog with other religions such as the Eastern Orthodoxy. I think that the Vatican likes to sometimes distance themselves from all other religions and faiths.