= Solution
Let $U_{(1)}=\min(U_1,U_2)$ and $U_{(2)}=\max(U_1,U_2)$ be the <order statistics>. For the symmetric <uniform distribution>, each observation is above or below $\theta$ with probability $1/2$, independently. Thus
$$
\mathbb P_\theta\{U_{(1)}\le\theta\le U_{(2)}\}
=1-(1/2)^2-(1/2)^2=1/2.
$$
So $\boxed{[U_{(1)},U_{(2)}]}$ is an exact 50% <confidence interval>.
The observed interval $[10,11.5]$ contains values impossible under the sampling model. Both observations must lie within one unit of the parameter, so the interval of all compatible values is $[U_{(2)}-1,U_{(1)}+1]=[10.5,11]$. It contains the true parameter almost surely. A <support-restricted confidence interval for a uniform location parameter> can therefore report
$$
\boxed{[10.5,11].}
$$
More precisely, the procedure $[U_{(1)},U_{(2)}]\cap[U_{(2)}-1,U_{(1)}+1]$ retains exact 50% unconditional coverage, since the second interval contains $\theta$ almost surely. The intersection is a nonempty interval for every sample in the model and equals $[10.5,11]$ for these data. Reporting the full compatible interval alone is also valid and has coverage one. On this sample's large-spread event it lies wholly between the observations; this does not turn the unconditional 50% procedure into a generally 100% procedure.
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