Find all the values of $a$ for which the equations
\begin{align*}
x^2 + ax + 1 &= 0, \\
x^2  - x - a &= 0
\end{align*}have a common real root.  Enter all the possible values, separated by commas.
Let $r$ be the common root, so
\begin{align*}
r^2 + ar + 1 &= 0, \\
r^2 - r - a &= 0.
\end{align*}Subtracting these equations, we get $ar + r + a + 1 = 0.$  This factors as $(r + 1)(a + 1) = 0,$ so $r = -1$ or $a = -1.$

If $r = -1,$ then $1 - a + 1 = 0,$ so $a = 2.$

If $a = -1,$ then $x^2 - x + 1 = 0,$ which has no real roots.

So the only possible value of $a$ is $\boxed{2}.$