1. **State the problem:** Find the exact value of $\csc\left(-\frac{5\pi}{6}\right)$.\n\n2. **Recall the definition:** $\csc(\theta) = \frac{1}{\sin(\theta)}$. So we need to find $\sin\left(-\frac{5\pi}{6}\right)$ first.\n\n3. **Use the odd property of sine:** $\sin(-\theta) = -\sin(\theta)$. Therefore, $\sin\left(-\frac{5\pi}{6}\right) = -\sin\left(\frac{5\pi}{6}\right)$.\n\n4. **Evaluate $\sin\left(\frac{5\pi}{6}\right)$:** $\frac{5\pi}{6}$ is in the second quadrant where sine is positive. $\sin\left(\frac{5\pi}{6}\right) = \sin\left(\pi - \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.\n\n5. **Calculate $\sin\left(-\frac{5\pi}{6}\right)$:** Using step 3, $\sin\left(-\frac{5\pi}{6}\right) = -\frac{1}{2}$.\n\n6. **Find $\csc\left(-\frac{5\pi}{6}\right)$:** $\csc\left(-\frac{5\pi}{6}\right) = \frac{1}{\sin\left(-\frac{5\pi}{6}\right)} = \frac{1}{-\frac{1}{2}} = -2$.\n\n**Final answer:** $\boxed{-2}$