多分これが一番早いと思います。-0と+0の安定ソートは出来ません。

stableSort.js

function stableSort(arr, callback){
  var indexMap = new Map(null);
  
  for(var i=0;i<arr.length;i++){
    indexMap.set(arr[i], i);
  }
  
  var result;
  arr.sort(function compare(a, b){
    result = (callback) ? callback(a, b) : false;
    if(result) return result;
    
    return indexMap.get(a) - indexMap.get(b);
  });
  
  return arr;
}

console.log([11,10,9,8,7,6,5,4,3,2,1,0].sort());      // [0, 1, 10, 11, 2, 3, 4, 5, 6, 7, 8, 9] (V8)
console.log(stableSort([11,10,9,8,7,6,5,4,3,2,1,0])); // [11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]

console.log([11,10,9,8,7,6,5,4,3,2,1,0].sort(_=> 0));        // [5, 11, 9, 8, 7, 6, 10, 4, 3, 2, 1, 0] (V8)
console.log(stableSort([11,10,9,8,7,6,5,4,3,2,1,0], _=> 0)); // [11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]

console.log([11,10,9,8,7,6,5,4,3,2,1,0].sort((a,b)=> a.toString()[0].localeCompare(b.toString()[0])));        // [0, 11, 1, 10, 2, 3, 4, 5, 6, 7, 8, 9] (V8)
console.log(stableSort([11,10,9,8,7,6,5,4,3,2,1,0], (a,b)=> a.toString()[0].localeCompare(b.toString()[0]))); // [0, 11, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9]