So, working on my theory that $? and $! are being trashed because of the call to the signal handler, I localised the variables $? and $! which has made a positive difference. Here is the code.

$SIG{ALRM} = sub {
    local $? = 0;
    local $! = 0;
    print( "Alarm triggered in $$\n" );
    unlink('/doesnt/exist');  # this will definitely fail
};

alarm(2);
my $rt = system("sleep 4");
print "system returned $rt, \$? is $?\n";
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And the output is -

Alarm triggered in 26346
system returned 0, $? is 0
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Give it a try in your code. Let me know how is goes.