  We put forward a new method for proving weak uniqueness of stochastic
equations with singular drifts driven by a non-Markov or infinite-dimensional
noise. We apply our method to study stochastic heat equation (SHE) driven by
Gaussian space-time white noise $$ \frac{\partial}{\partial t} u_t(x)=\frac12
\frac{\partial^2}{\partial x^2}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\,
x\in D\subset\mathbb{R}, $$ and multidimensional stochastic differential
equation (SDE) driven by fractional Brownian motion with the Hurst index
$H\in(0,1/2)$ $$ d X_t=b(X_t) dt +d B_t^H,\quad t>0. $$ In both cases $b$ is a
generalized function in the Besov space $\mathcal{B}^\alpha_{\infty,\infty}$,
$\alpha<0$. Well-known pathwise uniqueness results for these equations do not
cover the entire range of the parameter $\alpha$, for which weak existence
holds. What happens in the range where weak existence holds but pathwise
uniqueness is unknown has been an open problem. We settle this problem and show
that for SHE weak uniqueness holds for $\alpha>-3/2$, and for SDE it holds for
$\alpha>1/2-1/(2H)$; thus, in both cases, it holds in the entire desired range
of values of $\alpha$. This extends seminal results of Catellier and Gubinelli
(2016) and Gy\"ongy and Pardoux (1993) to the weak well-posedness setting. To
establish these results, we develop a new strategy, combining ideas from
ergodic theory (generalized couplings of Hairer-Mattingly-Kulik-Scheutzow) with
stochastic sewing of L\^e.
