  We study questions of existence and uniqueness of weak and strong solutions
for a one-sided Tanaka equation with constant drift \lambda. We observe a
dichotomy in terms of the values of the drift parameter: for \lambda\leq 0,
there exists a strong solution which is pathwise unique, thus also unique in
distribution; whereas for \lambda>0, the equation has a unique in distribution
weak solution, but no strong solution (and not even a weak solution that spends
zero time at the origin). We also show that strength and pathwise uniqueness
are restored to the equation via suitable "Brownian perturbations".
