Proposed resolution:

This wording is relative to N4981.

1. Modify [common.iterator] as indicated:

   namespace std {
     template<input_or_output_iterator I, sentinel_for<I> S>
       requires (!same_as<I, S> && copyable<I>)
     class common_iterator {
     public:
       […]
       template<classcommon_with<I> I2, sentinel_for<I> S2>
         requires sentinel_for<S, I2>
       friend constexpr bool operator==(
         const common_iterator& x, const common_iterator<I2, S2>& y);
       […]
     };
     […]
   }
   

2. Modify [common.iter.cmp] as indicated:

   template<classcommon_with<I> I2, sentinel_for<I> S2>
     requires sentinel_for<S, I2>
   friend constexpr bool operator==(
     const common_iterator& x, const common_iterator<I2, S2>& y);
   

   -1- Preconditions: x.v_.valueless_by_exception() and y.v_.valueless_by_exception() are each false.

   […]

common_iterator has the following equality operator:

template<class I2, sentinel_for<I> S2>
    requires sentinel_for<S, I2>
  friend constexpr bool operator==(
    const common_iterator& x, const common_iterator<I2, S2>& y);

which is quite useful when wrapping a C++20 input_iterator that does not model equality_comparable so that the quality operator required by the Cpp17InputIterator can still be synthesized.

However, the function signature does not check the correlation between I2 and I, which allows a common_iterator wrapping two completely unrelated iterators to validly compare (demo):

common_iterator<string::iterator, unreachable_sentinel_t> i1;
common_iterator<list<int>::iterator, unreachable_sentinel_t> i2;
i1 == i2; // unfortunately compile

The proposed resolution requires common_with<I, I2> to be satisfied to enhance semantics, which is also consistent with the signature of counted_iterator::operator==.