C++ visitor pattern handling templated string types?

Question

I'm trying to use the visitor pattern to serialize the contents of objects. However one snag I'm hitting is when I'm visiting strings. My strings are of a templated type, similar to STL's basic_string. So something like:

basic_string<char_type, memory_allocator, other_possible_stuff> \\ many variations possible!

Since I can have very many different templated string types, I can't go and add them to my visitor interface. It would be ridiculous. But I can't add templates to my VisitString method because C++ prevents using templates parameters in virtual methods.

So what are my options to work around this?

EDIT: I've added some basic code

class IVisitor
{
public: 
     virtual void VisitString(some_kind_of_string_type string) = 0; // this is what I want in theory
};

class MyObject
{
public:
    typedef basic_string<char8, myAllocator, some_flag> MyStringType;
    Accept(IVisitor* visitor)
    {
        visitor->VisitString(mString); 
    }
private:
   MyStringType string;
};

class MyOtherObject
{
public:
    typedef basic_string<char16, myOtherAllocator, some_other_flag> MyOtherStringType;
    Accept(IVisitor* visitor)
    {
        visitor->VisitString(mString); 
    }
private:
   MyOtherStringType string;
};


class Reader : public IVisitor
{ 
public:
    virtual void VisitString(some_kind_of_string_type string)
    {
         // read some data, give it to the string
    }
}

Answer 1

In the end, I went with a slightly different approach. Instead of hoping to use a visitor with templated methods (which is, of course, impossible), I decided to pass a visitor-like class as a template parameter to my object's visit method. Totally simplified example:

class SomeKindOfVisitor // doesn't need to derive from a base class. 
{
     template <class StringClass>
     void VisitString(StringClass& string) // I get to keep templated methods
}


class MyObject
{
typedef basic_string<char8, myAllocator, some_flag> MyStringType;

public:

   template <class VisitorClass>
   void Accept(VisitorClass& visitor)
   {
       vistior.VisitString<MyStringType>(mMyString);
   }
private:
    MyStringType mMyString;
}

With this method, I still get to use my templated strings while still being able to pass any kind of "visitor" to my objects.