Hi Karen,
Have a look at a family tree on Ancestry for Eugene AMEDEE and Bridget MOYLAN. They have Bridget's death cert which appears to confirm that she had a relationship with a person named POWER. It has not been uploaded so I suggest that you contact them or you can purchase it yourself. It seems to solve the 'Emerdy' name that appears on your family's certs.
This is how Bridget appears on Ancestry's indexes:
Death
Name: Bridt AMEDEE
Age: 74
Died: Mauritius, Victoria
Father: Pate MOYLAN
Mother: Hannah FOGARTY
Registration Year: 1909
Registration Number: 2561
Marriage
Name: Bridget MOYLEN
Spouse Name: Enge AMEDEE
Registration Year: 1873
Registration Number: 1737
This might be your Annie's birth:
Bridget MOYLAN
Father not named
Mother: Bridget MOYLAN
Birth Place: Melbourne, Victoria
Registration Year: 1864
Registration Number: 23689