It is certainly possible. However being possible does not necessarily mean that it is compatible. The question also covers quite a broad amount of information. It is possible to create a religion that completely revolves around science. If there isn't one already, then just go make one. Bam, you've got yourself a compatibility. However that brings into question what each person deems as compatible. You can certainly hold an irrational position by supporting both faith-based belief and simultaneously supporting a scientific method. Now when posing the question of "is this compatibility", you're going to get different answers depending on the person. Perhaps it is a compatibility, but probably not in the true sense. Things like catholicism, an extremist sect of islam, or judaism probably wouldn't reach a true compatibility depending on how they teach their views and reasoning behind their respective books. This is an important line to draw because someone can interpret each teaching in a completely different way than someone else. Again though, the large amounts of ambiguity in those teachings/texts may cause someone to say it isn't a true compatibility. Though, if you were to limit down each sect of every religion and have them give their teachings and interpretations to you in a straight and concise form, it would probably be easier to reach a decision. Just come to an agreement on what each party considers to be a compatibility and then you can start progressing through the issue. So as of the current moment, teh puma's opinion is that they hold the possibility of being compatible. However he sees no amount of sufficient reasoning behind the idea of them being compatible by his terms at this current point in time. So for right now, they aren't compatible.