One way of doing this would be to write a subclass of Template::Provider that records the location of the templates it fetches. For example:

package Recording::Provider;
use base qw(Template::Provider);

my $Template_name;
sub last_template { $Template_name };

sub fetch {
    my $self = shift;
    return( $self->SUPER::fetch($Template_name = shift) );
};
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You can then add this as the last provider in the chain - and the only one that handles the default template.

my @config = (
    INCLUDE_PATH    => '/Users/adrianh/Desktop/template/',
);

my $template = Template->new({
    EVAL_PERL => 1,
    LOAD_TEMPLATES => [
        Template::Provider->new(@config),
        Recording::Provider->new(@config, DEFAULT => 'default.t2'),
    ],
}) or die "could not make template\n";
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Your default.t2 template can then do something like this to show what template failed to load:

[% PERL %] print Recording::Provider->last_template [% END %] 
failed to load
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That last bit should probably more elegently implemented as a plugin - but I leave that as an exercise for the reader :-)