I don't think that Perl will expand ~ to mean /home/anonymous_monk/, so you will have to do this expansion yourself:

use lib '/home/anonymous_monk/perl5/lib/perl5';
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Alternatively see perlrun on how to set up @INC from the commandline:

export PERL5LIB=~/perl5/lib/perl5
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Also, see the local::lib documentation on how to really set up local::lib. There are some steps you need to perform after installing it to get things set up.