A median structure that can also remove its median

Design a structure over a stream of numbers that supports:

• add(x), insert a value.
• peek_median(), return the current median (average of the two middles when the count is even).
• pop_median(), remove and return the lower-middle element (for an odd count this is the unique median).

add(5); add(3); add(8)   # holds {3, 5, 8}, median 5
pop_median() -> 5        # now holds {3, 8}
peek_median() -> 5.5

Support add and pop_median in $O(\log n)$ and peek_median in $O(1)$.