Adding parentheses around $$arg[0] fixes that alright, but I don't care.

But you should care. If adding parenthesis helps, it's a parsing problem. Here's what B::Deparse has to say to the code:

sub test_a {
    use warnings;
    use strict;
    my $arg = shift();
    ref $arg ? $$arg->test_a([0]) : $arg;
}
sub test_b {
    use warnings;
    use strict;
    my $arg = shift();
    ref $arg ? test_a($$arg[0]) : $arg;
}
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So the call to test_a inside test_a is parsed as indirect method call syntax. Why? Because it's not predeclared. In Perl 5, a name only becomes visible in the statement after the declaration, which is why you can't write

my $sub = sub { ...; $sub->(); .. };
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So, in test_b you call test_a, which has already been declared. So either use parens after the function name, or predeclare it with

sub test_a;
sub test_a { ...; recurse into test_a here };
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