The variance inflation factor (VIF) for a variable in a regression model is given by:

$\text{VIF} = \frac{1}{1 - R_j^2}$

Where $R_j^2$ is the coefficient of determination from regressing variable $j$ on the other independent variables. In this case with two independent variables, $R_j^2$ equals the squared correlation coefficient between $X_1$ and $X_2$:

$R_j^2 = \rho_{X_1 X_2}^2$

Given that VIF = 15:

$\frac{1}{1 - \rho_{X_1 X_2}^2} = 15$

Solving for $\rho_{X_1 X_2}$:

$`$1` - \rho_{X_1 X_2}^2 = \frac{1}{15}$$

$\rho_{X_1 X_2}^2 = 1 - \frac{1}{15} = \frac{14}{15}$

$\rho_{X_1 X_2} = \sqrt{\frac{14}{15}} = \sqrt{0.9333} = 0.9661$

Therefore, the correlation coefficient between $X_1$ and $X_2$ that gives a VIF of 15 is 0.9661.