The meaning of compatibility is the ability of being comfortable with somebody, without any desires; just to feel free with someone.
Compatibility connects with various things like conversation, vision, hobbies and especially with the presence level of a person. It mostly depends on your presence level and the flow of your conversations. It is connected with Nakshatra also. Compatibility and zodiac are the leaves and Nakshatra is the root. Compatibility can be identified by your physche and your thought patterns. If you are in a calm flow than you cannot connect with a destructive flow person.
It is added with your zodiac signs also but nowadays, your psyche, presence level, thought patterns and the vision all matter. By knowing your and your partners behaviour and ways of life you can also be sure of your relationship. If you feel compatible with somebody, you don't have to say what to do or what not to. You will be naturally comfortable with each other.
Just to know about your compatibility you should share with me your thoughts and running matters of your mind.