Step 2 is to try the leftmost part 1 time (no TGA found).

Step 3 is to try the leftmost part 2 times (no TGA found).

Step 4 (and this is where the naive part goes bad) is to advance the leftmost starting point by one, since the match is unanchored.

can you elaborate more please? I cannot understand you. Because following your steps, the short string wouldn't match. Trying the leftmost part 2 times means ATCGTT = (\w\w\w)*? ok? 2 times. But how it is possible that it matches on CGTTGA, i cannot understand.

Here's another way to look at things: instrument the regex with  (?{ code }) (see Extended Patterns) print points to learn by experimentation. I'm also taking the liberty of introducing some other new constructs: the  (?:pattern) non-capturing grouping (also see Extended Patterns); the  /x regex modifier (all the preceding links found in perlre); and the  @- (aka @LAST_MATCH_START) array regex special variable (see perlvar).

First look at TGA matching against a simplified string without a  \G anchor. Note that in contrast to some other code examples in this thread, the beginning offset of a match is reported.

c:\@Work\Perl>perl -wMstrict -le
"my $s = 'XXXxxxTGAxxTGAxxxxxxx';
 while ($s =~ m{ (?{ print qq{trying a match at offset }, pos $s }) (?
+: \w\w\w)*? (TGA) }xmsg) {
     print qq{matched TGA beginning at offset $-[1]};
     }
"
trying a match at offset 0
matched TGA beginning at offset 6
trying a match at offset 9
trying a match at offset 10
trying a match at offset 11
matched TGA beginning at offset 11
[download]

After the successful TGA match at offsets 6 thru 8, the regex engine starts trying to match again at offset 9. The RE tries matches at offsets 9, 10 and 11 and finds a spurious (because it's not on a base-triplet boundary) match at offset 11-13. (I'm not sure why the RE doesn't try matching from offset 14 onward.)

Now consider the effect of adding a  \G anchor assertion.

c:\@Work\Perl>perl -wMstrict -le
"my $s = 'XXXxxxTGAxxTGAxxxxxxx';
 while ($s =~ m{ \G (?{ print qq{trying a match at offset }, pos $s })
+ (?: \w\w\w)*? (TGA) }xmsg) {
     print qq{matched TGA beginning at offset $-[1]};
     }
"
trying a match at offset 0
matched TGA beginning at offset 6
trying a match at offset 9
[download]

Now the RE can only begin another successful match at the offset immediately beyond the point at which the previous successful match ended, offset 9; it cannot try offsets 10 or 11 or any other because they do not satisfy the  \G assertion.

Supplemental: We just got finished saying that in

my $s = 'XXXxxxTGAxxTGAxxxxxxx';
while ($s =~ m{ (?{ print qq{trying a match at offset }, pos $s }) (?:
+ \w\w\w)*? (TGA) }xmsg) {
  print qq{matched TGA beginning at offset $-[1]};
  }
[download]

the RE will match the TGA at offset 11 because it's not constrained by a  \G assertion. So in

c:\@Work\Perl>perl -wMstrict -le
"my $s = 'XXXxxxTGAxxTGAxxxxTGAxx';
 while ($s =~ m{ (?{ print qq{trying a match at offset }, pos $s }) (?
+: \w\w\w)*? (TGA) }xmsg) {
   print qq{matched TGA beginning at offset $-[1]};
   }
"
trying a match at offset 0
matched TGA beginning at offset 6
trying a match at offset 9
matched TGA beginning at offset 18
[download]

(still no \G), why does the RE miss the TGA at offset 11 when there is another TGA present at offset 18 (which it does match)?

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Give a man a fish:  <%-{-{-{-<