The cake can be served right away or allowed to age, from three days to six weeks: Wrap the cake in cheesecloth, keep it at room temperature, and pour 2 to 3 tablespoons of rum over the top every few days, or once a week if storing longer.
Combine raisins, apricots, ginger, kumquats, star-anise pods, and vanilla-bean pod in a medium bowl. Bring water to boil in a small saucepan over high heat. Remove from heat, add tea, and cover; let steep 6 to 8 minutes. Strain brewed tea into bowl with fruit; let stand at room temperature at least 1 hour or up to overnight (the fruit will soften and absorb the liquid). Remove and discard star anise and vanilla pods.
Preheat oven to 375 degrees. Combine brown sugar, zest, and vanilla-bean seeds in a small bowl. Use your fingers to mix ingredients until well blended. In another bowl, lightly whisk together flour, baking powder, salt, ginger, nutmeg, cardamom, and fennel seed.
Butter an 8-inch square baking pan and line with parchment; butter parchment. With mixer on medium-high speed, beat butter and brown-sugar mixture until fluffy, about 3 minutes. Beat in eggs, one at a time, until incorporated. Add flour mixture and beat on medium-low speed until just combined. Add dried fruit mixture, pistachios, and marmalade; beat until just combined.
Transfer batter to prepared pan. Bake 15 minutes, then reduce oven temperature to 350 degrees and continue baking until cake is golden brown and a tester inserted in center comes out clean, 45 to 60 minutes. Using a toothpick or skewer, poke holes in top of cake; pour rum over cake. Let cool completely in pan.
In a small bowl, stir together confectioners' sugar and lemon juice until smooth. Drizzle icing over cake. Garnish with thin slices of fruit from additional marmalade.