Thanks, gfvalvo.
I tried your suggestion and prepended "extern" to the variable definitions in multi_file.h. That did not remedy the problem. Here is the code for the updated file.
/* FILE: multi_file.h */
#ifndef multi_file_h
#define multi_file_h
#include "Arduino.h"
extern byte var_1 = 0;
void fnc_1( void );
extern byte var_2 = 0;
void fnc_2( void );
#endif