Common code between unrelated classes

In a Client Server setup execution flow is often akin to:

a) Read Client Data -> b) Process -> c) Write Server Data
d) Read Server Data -> e) Update -> f) Write Client Data

On the Client side, I'm faced with two classes.

class Base_Message { virtual ~Base_Message() {} };
class CMsg1 : public Base_Message {}; //CMsg: from Client
class SMsg1 : public Base_Message {}; //SMsg: to Client

class Handle_Incoming : public Base_Handler {
CMsg1 Client_Msg ;
public:
void Process () { /* work */ }
};
class Handle_Outgoing : public Base_Handler {
SMsg1 Server_Msg ;
public:
void Update ( /*...*/ ) { /*update Server_Msg* pre f) above */ }
};

Incoming client messages are local to the Handle_Incoming class.
Outgoing server data is local to the Handle_Outgoing. The outgoing
server msg (SMsg1) needs to be shared between the Handle_Incoming and
Handle_Outgoing classes. I can hear you saying, "Make a base class and
derive from it!", and yes, I could do that. In fact I did that as a
temporary measure, but this defines a relationship between the
Handle_Incoming and Handle_Outgoing classes that doesn't exist.

In my view using an interface will not help in this situation, because
it doesn't offer implementation sharing. Inheritance offers a
relationship with implementation sharing, an interface offers a
relationship without implementation sharing, but I want implementation
sharing without the relationship.

My question becomes: How do I make SMsg1 available for both
Handle_Incoming and Handle_Outgoing classes when a relationship among
them doesn't exist?

Thanks in advance.

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