In this paper we enumerate the centrally symmetric lozenge tilings of a hexagon with a fern removed from its center. The proof is based on a variant of Kuo's graphical condensation method. An unexpected connection with the total number of tilings is established~---~when suitably normalized, the number of centrally symmetric tilings is equal to the square root of the total number of tilings. The results we present can be regarded as a new extension of the enumeration of self-complementary plane partitions that fit in a box.