Process the strawberries with salt in a blender or food processor until a chunky puree is formed. Set aside.
In a medium mixing bowl, combine the eggs and sugar and beat until slightly thickened and pale yellow. Set aside.
Bring half and half to a simmer in a heavy medium saucepan. In a slow, steady stream, beat the hot half and half into the eggs. Pour the egg mixture back into the pan and place over low heat, stirring constantly with a wooden spoon or whisk until mixture thickens. Don't allow the mixture to boil, otherwise the eggs will scramble.
Remove from heat. Set a strainer over a clean bowl and pour the egg mixture through the sieve to remove any stray pieces of egg. Allow the custard to cool slightly and stir in the heavy cream, rum and vanilla extract. Add the strawberry puree and stir to combine.
Cover the custard with plastic wrap, resting the plastic directly on the custard so that a skin doesn't form as it cools. Refrigerate several hours or overnight.
Stir the chilled custard then freeze it in your ice cream machine according to the manufacturers instructions. When finished, the ice cream will be soft, but ready to eat. For firmer ice cream, transfer to a freezer safe container and freeze for at least two hours.