This is a drink which can be nice with all fresh and natural ingredients. Make your own grenadine (using freshly processed pomegranate juice which is magenta in colour, please don't get the bottled ones which are dark red in colour), fresh lemon juice and a cheat's homemade ginger ale (which is fresh ginger juice syrup and soda). The fruitiness of pomegranate plays wonderfully with spicy ginger.
Chris, I used an honest-to-goodness homemade grenadine - a recipe I love and have made for years - and the taste of it in the ginger ale (Fever Tree Refreshingly Light) threw me off. But I agree that it might be best to go all homemade with this. I'll try your ginger ale suggestion.
While I can see how even my take is polarizing, looking at Difford's tasting notes of Fever Tree Refreshingly Light Ginger Ale, makes me think it could just be that. I make my ginger syrup using Jeffrey Morgenthaler's recipe of equal parts ginger, water and sugar by weight, resulting in a syrup which is balanced but still spicy. I would also suggest trying it with a ginger 'beer' as well. I feel like spicy ginger works well with the lemon to calm the sweet grenadine down. But one more warning, I'm only in my 20s, and I have a wet palate.