So I understand that you may have two equal objects but they are not the same instance.
Such as
Integer a = new Integer(3);
Integer b = new Integer(3);
In which case a.equals(b) because they refer to the same intrinsic value but a != b because they are two different objects.
There are other implementations of Set, such as IdentitySet, which do a different comparison between items.
However, I think that you are trying to apply a different philosophy to Java. If your objects are equal (a.equals(b)) although a and b have a different state or meaning, there is something wrong here. You may want to split that class into two or more semantic classes which implement a common interface - or maybe reconsider .equals and .hashCode.
If you have Joshua Bloch's Effective Java, have a look at the chapters called "Obey the general contract when overriding equals" and "Minimize mutability".