  Let $U$ be a connected, simply connected compact Lie group with
complexification $G$. Let $\mathfrak{u}$ and $\mathfrak{g}$ be the associated
Lie algebras. Let $\Gamma$ be the Dynkin diagram of $\mathfrak{g}$ with
underlying set $I$, and let $U_q(\mathfrak{u})$ be the associated quantized
universal enveloping $*$-algebra of $\mathfrak{u}$ for some $0<q$ distinct from
$1$. Let $\mathcal{O}_q(U)$ be the coquasitriangular quantized function Hopf
$*$-algebra of $U$, whose Drinfeld double $\mathcal{O}_q(G_{\mathbb{R}})$ we
view as the quantized function $*$-algebra of $G$ considered as a real
algebraic group. We show how the datum $\nu = (\tau,\epsilon)$ of an involution
$\tau$ of $\Gamma$ and a $\tau$-invariant function $\epsilon: I \rightarrow
\mathbb{R}$ can be used to deform $\mathcal{O}_q(G_{\mathbb{R}})$ into a
$*$-algebra $\mathcal{O}_q^{\nu,\mathrm{id}}(G_{\mathbb{R}})$ by a modification
of the Drinfeld double construction. We then show how, by a generalized theory
of universal $K$-matrices, a specific $*$-subalgebra
$\mathcal{O}_q(G_{\nu}\backslash \backslash G_{\mathbb{R}})$ of
$\mathcal{O}_q^{\nu,\mathrm{id}}(G_{\mathbb{R}})$ admits $*$-homomorphisms into
both $U_q(\mathfrak{u})$ and $\mathcal{O}_q(U)$, the images being coideal
$*$-subalgebras of respectively $U_q(\mathfrak{u})$ and $\mathcal{O}_q(U)$. We
illustrate the theory by showing that two main classes of examples arise by
such coideals, namely quantum flag manifolds and quantum symmetric spaces
(except possibly for certain exceptional cases). In the former case this
connects to work of the first author and Neshveyev, while for the latter case
we heavily rely on recent results of Balagovi\'{c} and Kolb.
