Positive correlation, negative coefficient

Idea. In the joint regression, the coefficient on $x_2$ measures what $x_2$ says about $y$ once $x_1$ is known. Here $x_2$ is mostly a noisy copy of $x_1$, and it correlates with $y$ less than that copy would, so its coefficient turns negative.

Step 1: the normal equations. $$\begin{array}{rl}{b}_1+0.6b_2& =0.4,\\ 0.6b_1+b_2& =\mathrm{0.2.}\end{array}$$

Step 2: solve. $$b_2=\frac{0.2-0.4\cdot 0.6}{1-0.6^2}=\frac{-0.04}{0.64}=-\frac{1}{16},$$ and $b_1=0.4-0.6b_2=0.4375$.

Why the sign flips. Through $x_1$ alone, $x_2$ would be expected to correlate with $y$ at $0.6\times 0.4=0.24$. Its actual correlation, $0.2$, is lower, so the part of $x_2$ not explained by $x_1$ is negatively related to $y$. In general the sign of $b_2$ is the sign of $r_{y2}-r_{y1}{r}_{12}$.

The lesson. A coefficient in a multiple regression is not a correlation. A signal with a positive correlation to the target can get a negative weight when combined with a stronger, correlated signal.