The flaw in the mistranscription theory is that this Margaret's surname is Cohen in both the St Nicholas, Liverpool, parish record, and also the BMD marriage index. So either it's the marriage of a different William Birch, or else, if it's the right William Birch marriage, then Margaret from the IOM was actually called Cohen or had intentionally changed the spelling of her name.

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MARGARET COHEN Pedigree
Female Family

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Event(s):
Birth:
Christening:
Death:
Burial:

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Marriages:
Spouse: WILLIAM BIRCH Family
Marriage: 29 JUL 1860 Saint Nicholas, Liverpool, Lancashire, England

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Messages:
Extracted marriage record for locality listed in the record. The source records are usually arranged chronologically by the marriage date.

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England & Wales, FreeBMD Marriage Index: 1837-1915
about William Birch
Name: William Birch
Year of Registration: 1860
Quarter of Registration: Jul-Aug-Sep
District: Liverpool (1837-1934)
County: Lancashire
Volume: 8b
Page: 329

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England & Wales, FreeBMD Marriage Index: 1837-1915
about Margaret Cohen
Name: Margaret Cohen
Year of Registration: 1860
Quarter of Registration: Jul-Aug-Sep
District: Liverpool (1837-1934)
County: Lancashire
Volume: 8b
Page: 329

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Jean C