In this paper, we establish a structure theorem for projective Kawamata log terminal (klt) pairs (X, ∆) with nef anti-log canonical divisor; specifically, we prove that up to replacing X with a finite quasi-étale cover, X admits a locally trivial rationally connected fibration onto a projective klt variety with numerically trivial canonical divisor. Our structure theorem generalizes previous works for smooth projective varieties and reduces the structure problem to the singular Beauville-Bogomolov decomposition for Calabi-Yau varieties. As an application, the projective varieties of klt Calabi-Yau type, which naturally appear as an outcome of the Log Minimal Model Program, are decomposed into building block varieties, namely, rationally connected varieties and Calabi-Yau varieties. SHIN-ICHI MATSUMURA AND JUANYONG WANG 4.2. Case of Q-factorial terminal pairs with splitting tangent sheaf 40 4.3. Case of klt pairs with nef anti-log canonical divisor 46 References 51