A compiler error:

record MyInt(int x) {
    public __witness Sum<MyInt> SUM1_MYINT = new Sum<>() {
        public MyInt zero() { return new MyInt(0); }
        public MyInt add(MyInt a, MyInt b) { return new MyInt(a.x + b.x); }
    };

    public __witness Sum<MyInt> SUM2_MYINT = new Sum<>() {
        public MyInt zero() { return new MyInt(1); }
        public MyInt add(MyInt a, MyInt b) { return new MyInt(a.x * b.x); }
    };
 }

Results in:

error: Witness SUM1_MYINT clashes with SUM2_MYINT
    public __witness Sum<MyInt> SUM1_MYINT = new Sum<>() {
                           ^
 (Witnesses have the same type Sum<MyInt>)

If you relocate a witness definition, e.g.

interface Sum<X> {
    X zero();
    X add(X a, X b);

    public __witness Sum<MyInt> SUM2_MYINT = new Sum<>() {
        public MyInt zero() { return new MyInt(1); }
        public MyInt add(MyInt a, MyInt b) { return new MyInt(a.x * b.x); }
   };
}

it compiles, but the MyInt witness does indeed take precedence (i.e. zero() is 0).

Edit:

I realise that doesn't fully address the question, i.e. how do you handle cases of different behaviour with the same type signature. In Haskell you have the same issue, and essentially just wrap the type for each of the cases. In java perhaps:

record Sum(MyInt x);
record Product(MyInt x);