This is that flexibility thing that Scala folks always talk about, right? ;) (aka I finally decided to learn about TypeFamilies, then had some fun)

gistfile1.scala

{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE RebindableSyntax #-}

module Foo where

import Prelude hiding ((+), Num(..))
import qualified Prelude as P

default (Integer)

class Add a b where
  type AddThings a b
  (+) :: a -> b -> AddThings a b

instance Add String Integer where
  type AddThings String Integer = String
  a + b = a ++ show b

instance Add Integer String where
  type AddThings Integer String = String
  a + b = show a ++ b

instance P.Num a => Add a a where
  type AddThings a a = a
  a + b = a P.+ b

fromInteger :: Integer -> Integer
fromInteger = id

-- λ> :set -XRebindableSyntax
-- λ> :l /tmp/foo.hs
--
-- λ> 3 + 4
-- 7
--
-- λ> 3 + "four"
-- "3four"
--
-- λ> "three" + 4
-- "three4"