REAMS – Local of ‘Rheims’ the surname has crossed over from Norfolk to Lincoln. The spelling is much corrupted.
Hugo de Reymes co of Norf. 1773. A. Victor de Reymes, 34 Hen 111 FF viii 143.
William de Reymes co. Suff. Edw. I. R.
1653 married – William Reymes and Dorothy Fowle : St David Backchurch, MDB co. Linc.

From the English and Welsh Surname Dictionary as shown at Ancestry.com.