  We constrain the level of polarized anomalous microwave emission (AME) on
large angular scales using $\textit{Planck}$ LFI and $\textit{WMAP}$
polarization data within a Bayesian CMB analysis framework. We model
synchrotron emission with a power-law spectral energy distribution, and the sum
of AME and thermal dust emission through linear regression with the
$\textit{Planck}$ HFI 353 GHz data. This template-based dust emission model
allows us to constrain the level of polarized AME while making minimal
assumptions on its frequency dependence. We neglect cosmic microwave background
fluctuations, but show through simulations that these have a minor impact on
the results. We find that the resulting AME polarization fraction confidence
limit is sensitive to the polarized synchrotron spectral index prior, and for
priors steeper than $\beta_{\mathrm{s}} = -3.1\pm0.1$ we find an upper limit of
$p_{\mathrm{AME}}^{\rm max}\lesssim 0.6\,\%$ ($95\,\%$ confidence). In
contrast, for $\beta_{\mathrm{s}}=-3.0\pm0.1$, we find a nominal detection of
$p_{\mathrm{AME}}=2.5\pm1.0\,\%$ ($95\,\%$ confidence). These data are thus not
strong enough to simultaneously and robustly constrain both polarized
synchrotron emission and AME, and our main result is therefore a constraint on
the AME polarization fraction explicitly as a function of $\beta_\mathrm{s}$.
Combining the current $\textit{Planck}$ and $\textit{WMAP}$ observations with
measurements from high-sensitivity low-frequency experiments such as C-BASS and
QUIJOTE will be critical to improve these limits further.
