I agree that pronunciations, and sometimes spellings (inverting a final "-er" to "-re", e.g.) complicate this type of puzzle.  It is hard to communicate in writing how we pronounce.  My dictionary (an American product) shows the same diacriticals for "sear" (v.,to parch) and "seer" (n.,a wise guy).  Most of our differences would be placing accents differently in multisyllabic words (not sure whether that affects status as homophones).  Probably I should have used brian's term, since while homonym gives homophone as a synonym, it also may mean homograph or namesake.  But homophones also may refer to either words or syllables.

Is there any way to indicate to me how you would pronounce "sear" and "seer" -- e.g. different rhymes for each? 

I grant that US pronunciation of "Mary" (proper name), and "Merry" (gay, also a proper name), and "marry" (nuptials) are pronounced alike (not sure where UK differs).  If one were calling Mary or merry, it might be with a strong accent on the first syllable (MAAAAYre).  Perhaps there are as many differences within our respective countries.

The university here has been publishing a "Dictionary of American Regional English" over decades: at last look it had five hefty volumes (as I recall), and not sure that completed the work -- our closest approximation to the OED, though with a more specialized scope.