Thanks to all. The solution which worked (avoiding the MyMod::MYMOD_DEFAULT redefined at /usr/local/lib/perl5/5.8.7/constant.pm message and allowing the constant to be seen "above" and "below" the module which defined it was:

fred.pl
use MyMod;
MyMod::db->fred(MYMOD_DEFAULT);
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MyMod.pm
use strict;
use warnings;
package MyMod;

use constant MYMOD_DEFAULT => 1;
use Exporter qw(import);
our @EXPORT = qw (MYMOD_DEFAULT);
# changing the use to require did the trick:
require MyMod::db;
sub dave {
        print "The constant is ", $_[1], "\n";
}
1;
[download]

MyMod/db.pm
use strict;
use warnings;
package MyMod::db;
use MyMod;

sub fred {
        print "hello\n" if (MYMOD_DEFAULT);
}
1;
[download]