If it isn't already, blind-bake your pastry case: Line a 23cm tart tin with pastry, prick the bottom all over with a fork, cover with parchment and fill with dried beans. Bake at 180 C / Gas 4 for about 30 minutes, or till golden. Remove the beans and parchment and let the pastry case cool.
For the tart, preheat the oven to 170 C / Gas 3.
Zest one lime, then juice all of the limes.
Combine the lime juice and zest with the eggs, egg yolks, cream, ginger and sugar. Whisk till well combined, then pour into cooled pastry case.
Bake in the preheated oven for 45 minutes, or until set. Remove from oven and let cool slightly.
Meanwhile, heat the lime preserve over low heat till melted. Remove from heat and let cool slightly.
Top the still-warm tart with the warm lime preserve. Spread over the tart evenly. Allow to cool completely before slicing and serving.