In a large bowl or the bowl of a stand mixer, combine flour, corn starch, baking powder, salt, and sugar, and mix on low speed until ingredients are distributed evenly.
Add butter and mix until completely combined.
Add oil and coconut milk and mix until completely combined.
Add eggs one at a time, mixing between each addition until just combined.
Add rum and mix until just combined.
Pour into a 12 cup greased and floured Bundt pan and bake for 55 minutes. If a toothpick inserted into the batter comes out moist, bake for an additional five minutes and test again.
Let cake cool in the Bundt pan while you prepare the syrup. (It shouldn't stick to the pan if you greased it well, but if it seems stuck, wait five minutes, turn it out, and then return it to the Bundt pan before proceeding with adding the syrup to the cake.)
Syrup Instructions
Add all ingredients to a medium saucepan over medium heat.
Once the mixture is boiling, reduce to a simmer and cook for 10 minutes before removing from heat.
Poke holes deep into the cake all over (the cake should still be in the Bundt pan) and slowly and evenly pour all but 1/4 cup of the syrup onto the cake.
Cover cake with foil or plastic wrap and wait a minimum of two hours for the cake to cool and absorb the syrup. The cake tastes better one or two days after you make it, so it's fine to let it sit overnight or longer.
Turn out onto a plate or cake pedestal.
Stir the remaining rum syrup as it may have separated a little bit. Drizzle all over the top of the cake and serve.
Notes
This cake gets better with age. Store covered at room temperature for up to one week.