Independence is state of mind which you achieve when you are not dependent financially, legally, personally etc on client. Now you dont have shares etc of client , you dont have contract of other prohibited services with client, you dont have family relationship with client, so you are not under influence of client that means you are having independence of Mind.
but if you party regularly with client, celebrate all festivals together and appear very close to each other in public, then public will get confused are you really independent. This is not independence in appearance. So you should ensure that you appear independent along with independence of mind.