I don't know why it's happening, but localizing the alarm handler in Enginesub.pm seems to solve it over here.

sub go {

    my $self = shift @_;
    local $SIG{ALRM} = \&ti;

    eval {
...
[download]

Output:

5 seconds main input:
no input running....
name name1 (2 seconds) sub input:
no input running defaults.....
name name2 (2 seconds) sub input:
no input running defaults.....
5 seconds main input:
no input running....
name name1 (2 seconds) sub input:
no input running defaults.....
name name2 (2 seconds) sub input:
no input running defaults.....
[download]

Update: Aha, of course... Since you don't localize the handler in Enginesub, it will replace the one you had for your main loop. This means that the second time the 5 second timeout triggers, it will call the ti sub instead of the timed_out sub. This in turn means that your if ($@ =~ /BLAH/) is false, since ti sets it to TIMEOUT instead.

Changing your main while loop to:

while() {

    eval {
        alarm(5);
        print "\n\n5 seconds main input: ";
        chomp(my $input = <STDIN>);
        alarm(0);
    };

    if ($@ =~ /BLAH/) {
        print "\n\nno input running....\n\n";
        $engine->run();
    }

    if ($@ =~ /TIMEOUT/) {
        print "\n\nOops, still using ti handler...\n\n";
    }

    sub timed_out {
        die "BLAH";
    }
}
[download]

The output is:

5 seconds main input:
no input running....
name name1 (2 seconds) sub input:
no input running defaults.....
name name2 (2 seconds) sub input:
no input running defaults.....
Oops, still using ti handler...
5 seconds main input:
Oops, still using ti handler...
5 seconds main input:
Oops, still using ti handler...
[download]

(Removed some line breaks to make things easier to read)