Add the canned lychees to a tall highball glass and muddle firmly to break them down.
Add the mint leaves and press lightly once or twice to release the oils.
Add the rum, lychee syrup, simple syrup or rose syrup, and lime juice.
Stir briefly to combine.
Fill the glass with ice.
Top with soda water and stir very gently.
Garnish with a fresh mint sprig and a lychee on a cocktail skewer.
Notes
The canned lychee syrup adds floral flavor but isn't as sweet as simple syrup - top it up with a small amount of simple syrup to get the right balance.