  We introduce a new class of reflected backward stochastic differential
equations with two c\`adl\`ag barriers, which need not satisfy any separation
conditions. For that reason, in general, the solutions are not semimartingales.
We prove existence, uniqueness and approximation results for solutions of
equations defined on general filtered probability spaces. Applications to
Dynkin games and variational inequalities, both stationary and evolutionary,
are given.
