  We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$,
and $F_i$ on shifted skew semistandard tableaux. We show that the primed
operators and unprimed operators each independently form type A Kashiwara
crystals (but not Stembridge crystals) on the same underlying set and with the
same weight functions. When taken together, the result is a new kind of
`doubled crystal' structure that recovers the combinatorics of type B Schubert
calculus: the highest-weight elements of our crystals are precisely the shifted
Littlewood-Richardson tableaux, and their generating functions are the (skew)
Schur $Q$-functions. We give local axioms for these crystals, which closely
resemble the Stembridge axioms for type A. Finally, we give a new criterion for
such tableaux to be ballot.
