# awk '!/^ *$/{if(w==1){x1=xy;x2=$1}else{if($1)x1=$1;getline;if($0)x2=$1; else next};if(x1~/^A_/){if(x2~/^C_/){w=0;print x1 FS x2}else{w=1;xy=x2}}
}' file >A_C_pairs
# awk '/^A/{a[x++]=$1}END{for(i=0;i<x;i++)print a[i]}' file >A
# awk '/^C/{a[x++]=$1}END{for(i=0;i<x;i++)print a[i]}' file >C

# cat A
A_ABCD_13208
A_ABCDEF715
A_DBCDI_1353
A_EFGH_9099
A_QRST_9938
A_XVYZ_24808
A_TRST_7849
A_ABCDI_15931
A_ABCDI_15930
A_GHJK_30564
A_STDT_2657

# cat C
C_ABCD_13208
C_DBCDI_1353
C_EFGH_24808
C_QRST_9938
C_GHIH_9099
C_TRST_7849
C_GHJK_30564
C_STDT_2657

# cat A_C_pairs
A_ABCD_13208 C_ABCD_13208
A_ABCDEF715 C_DBCDI_1353
A_DBCDI_1353 C_EFGH_24808
A_EFGH_9099 C_QRST_9938
A_XVYZ_24808 C_GHIH_9099
A_ABCDI_15930 C_GHJK_30564
A_GHJK_30564 C_STDT_2657