Via Unix join:

% join -t $'\t' -11 -21 -o 1.2,1.3,1.4,2.2,2.3,2.4,2.5 <(awk -v FS="\t" -v OFS="\t" '{ print $1"_"$2, $0 }' A.txt | sort -k1) <(awk -v FS="\t" -v OFS="\t" '{ print $1"_"$2, $0 }' B.txt | sort -k1)          
chr1    9133639 T   chr1    9133639 T   GI_02334
chr2    6134363 C   chr2    6134363 C   GI_02338
chr4    6344639 A   chr4    6344639 A   GI_02365

This works by creating a unique identifier key from the chromosome and position of each file with awk, sorting on that key, and joining the two files on that key (-11 -21). The -o 1.2,1.3,1.4,2.2,2.3,2.4,2.5 option prints out non-key columns.

Using sorted data will add runtime, but it may be preferable to reading both files into associative arrays in awk (or dictionaries in Python, hash tables in Perl, etc.) as the latter approach can use a lot of memory, which could be troublesome if your second file is very large.

Here's a dictionary approach in Python, using a tuple of the first and second column values as a key:

#!/usr/bin/env python

import sys

f1_fn = "A.txt"
f2_fn = "B.txt"

f1 = {}
with open(f1_fn, 'r') as f1_fh:
    for l in f1_fh:
        e = l.rstrip().split('\t')
        f1[(e[0], e[1])] = e

f2 = {}
with open(f2_fn, 'r') as f2_fh:
    for l in f2_fh:
        e = l.rstrip().split('\t')
        f2[(e[0], e[1])] = e

# inner join
k1 = f1.keys()
k2 = f2.keys()
for k in k1:
    if k in k2:
        sys.stdout.write('{}\n'.format('\t'.join(f1[k] + f2[k])))