= Solution
The <Tanaka equation> is an example:
$$
\boxed{dX_t=s(X_t)\,dB_t,\quad X_0=0,\qquad s(x)=\begin{cases}1&x\geq0,\\-1&x<0.\end{cases}}
$$
It has <weak solutions> and <uniqueness in law>, but lacks <pathwise uniqueness>. The value $s(0)=1$ is deliberate: using the <sign function> with value zero at zero would admit the identically zero solution and would change the example.
For a quick verification, take a <Brownian motion> $W$ and put $B=\int s(W)dW$. Its <quadratic variation> is $t$, so it is a <Brownian motion> by the <Lévy characterization of Brownian motion>, and <associativity of stochastic integration> gives $W=\int s(W)dB$. Every <weak solution> likewise has <quadratic variation> $t$ and therefore the law of <Brownian motion>. On this same enlarged <filtration>, both $W$ and $-W$ solve the equation driven by $B$: $s(-W)=-s(W)$ away from zero, and the <Brownian zero set> has zero time measure, so the discrepancy at zero contributes nothing to the <Itô integral>. These solutions differ with positive probability at any positive time. This proves the claimed distinction without relying on a choice of zero convention left unstated.
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