Create a type that's a subtype of Str. For that type, create a coercion rule so that Undef is coerced to the empty string. Use that type and coerce=>1 for the attribute.

use Moose::Util::TypeConstraints;

subtype 'MyStr',
   as 'Str';

coerce 'MyStr',
   from 'Undef',
      via { '' };

no Moose::Util::TypeConstraints;

has foo => (
   isa    => 'MyStr',
   coerce => 1,
   ...
);
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Can't think of a good name for it.