C++14, 118 108 bytes

Using the algorithm from w0lf's Haskell answer.

As unnamed generic lambda. First parameter is a container of the values to dropsort (like vector<int>) and second parameter requires a compatible empty container of containers (like vector<vector<int>>) for the return value via reference.

In the first version of the program, there was R.clear;() as first statement, so that the container of containers didnt need to be empty. Peter Cordes thought this could into the specification, so dropping 10 byte for that.

[](auto A,auto&R){for(auto x:A){for(auto&D:R)if(D.back()<x){D.push_back(x);goto F;}R.emplace_back(1,x);F:;}}

Try it online!

Ungolfed:

[](auto A,auto&R){
 for(auto x:A){       //foreach item
  for(auto&D:R)       //foreach result list
   if(D.back()<x){    //x bigger than last element
    D.push_back(x);   //add x
    goto F;           //break and jump over the emplace
   }
  R.emplace_back(1,x);//create new list with this element
  F:;
 }
}