Let $(X,\mathcal{O}_X)$ be a quasi-compact ringed space which is locally isomorphic to an affine variety.
Is it true then that $(X,\mathcal{O}_X)$ is always locally isomorphic to an irreducible affine variety?
Intuitively I'd like to say yes, if we take $x\in X$ then we have some open $U\subseteq X$ with $x\in U$ and $(U,\mathcal{O}_X)\cong(V,\mathcal{O}_V)$ for some affine $V$ via an isomorphism $\varphi$. We can then find some closed irreducible $W\subseteq V$ with $\varphi(x)\in W$, but then $\varphi^{-1}(W)$ is not necessarily open in $X$, and I can't seem to find a useful open set related to it.
However on the other hand I'm struggling to come up with a counterexample, any help would be much appreciated.
As in my comment, the answer is no.
If we take our space to be $V(XY)$ over some field $k$, this will be two lines meeting only at $(0,0)$. Taking $x=(0,0)$, any open set containing $x$ will just be $V(XY)$ minus a finite set of points. If this open set is isomorphic via a map $\psi$ to some affine $W$, then we would have $W=\psi(V(X))\cup\psi(V(Y))$. These sets are both closed, non-empty, and neither can be equal to the whole of $W$, and so $W$ cannot be irreducible.