1. **State the problem:** Calculate $\tan 150^\circ - \sin 135^\circ$.\n\n2. **Recall the formulas and values:**\n- $\tan(180^\circ - \theta) = -\tan \theta$.\n- $\sin(180^\circ - \theta) = \sin \theta$.\n- Common angle values: $\tan 30^\circ = \frac{1}{\sqrt{3}}$, $\sin 45^\circ = \frac{\sqrt{2}}{2}$.\n\n3. **Evaluate each term:**\n- $\tan 150^\circ = \tan(180^\circ - 30^\circ) = -\tan 30^\circ = -\frac{1}{\sqrt{3}}$.\n- $\sin 135^\circ = \sin(180^\circ - 45^\circ) = \sin 45^\circ = \frac{\sqrt{2}}{2}$.\n\n4. **Substitute values:**\n$$\tan 150^\circ - \sin 135^\circ = -\frac{1}{\sqrt{3}} - \frac{\sqrt{2}}{2}.$$\n\n5. **Find common denominator and combine:**\nCommon denominator is $2\sqrt{3}$.\n$$-\frac{1}{\sqrt{3}} = -\frac{2}{2\sqrt{3}}, \quad \frac{\sqrt{2}}{2} = \frac{\sqrt{2}\sqrt{3}}{2\sqrt{3}} = \frac{\sqrt{6}}{2\sqrt{3}}.$$\n\n6. **Rewrite expression:**\n$$-\frac{2}{2\sqrt{3}} - \frac{\sqrt{6}}{2\sqrt{3}} = \frac{-2 - \sqrt{6}}{2\sqrt{3}}.$$\n\n7. **Rationalize denominator:**\n$$\frac{-2 - \sqrt{6}}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{(-2 - \sqrt{6})\sqrt{3}}{2 \times 3} = \frac{(-2 - \sqrt{6})\sqrt{3}}{6}.$$\n\n**Final answer:**\n$$\boxed{\frac{(-2 - \sqrt{6})\sqrt{3}}{6}}.$$