The spelling of /g/ is <g> or <gg>, but we also see <gu> in rogue. The /Ɵ/ sound is spelled <th> throughout.
This is a contrast between a voiced stop and a voiceless fricative, well separated in the mouth. It does not cause any problems.
I was tempted to include the pair egg/Eth, though it was not in the dictionary, but Eth (short for Ethel) is now an unusual name, remembered chiefly from the character Eth Glum in the 1950s radio show Take it from Here. Other interesting pairs include:
The figure for the mean density is 1.9%. The set makes 43 semantic contrasts, a loading of 62%.
Aga Arthur auger author augers authors beg Beth berg berth bergs berths berg birth bergs births boggy bothy clog cloth clogs cloths digger dither diggers dithers Doug doth dug doth eager aether eager ether erg earth frog froth frogs froths gain thane gains thanes gaud thawed gauze thaws Gertie thirty ghat thought ghats thoughts gird third girds thirds goal thole goals tholes gong thong gongs thongs gore thaw gore Thor grace Thrace grill thrill grilled thrilled grills thrills grilling thrilling groan throne groans thrones groat throat groats throats grove throve grow throe grows throes grow throw growing throwing grows throws grew threw grown thrown grown throne gum thumb gummed thumbed gumming thumbing gums thumbs guy thigh guys thighs hag hath legal lethal overgrown overthrown pig pith piggy pithy piggier pithier piggiest pithiest plaguing plaything rogue wroth Whig with Whig withe wig with wig withe wigs withes
John Higgins, Shaftesbury, September 2010.