Cut each lime into eight wedges. Set four wedges aside for serving and add the remaining wedges to a large glass measuring cup or small pitcher along with the strawberries, cucumber, mint, and sugar. Using a cocktail muddler or the end of a wooden spoon, mash and muddle everything together until jammy and fragrant and the sugar has mostly dissolved into the mixture. Add in the rum and stir until well combined and the sugar has dissolved.
Half-fill four highball glasses with ice and pour over the rum mixture. Top with soda water and garnish with the remaining lime wedges. Enjoy!