Network densification is an emerging technology to achieve the key performance indicators expected in future wireless systems. However, several practical challenges as the appearance of interference channels need to be addressed. Interference alignment (IA) is a powerful technique to manage multiple concurrent transmissions in ultra-dense networks (UDNs). The fully digital (FD) IA has been widely studied. However, the research on IA with hybrid digital-analog beamforming (HB) for massive multiple-input multiple-output (mMIMO) scenarios, is limited. The reported works on HB-IA are designed assuming ideal phase shifters (PSs) without quantized phase constraints. Therefore, their performance severely degrades in practical systems with low-resolution PSs. Although quantized phase restrictions were considered in some HB schemes, they were limited to scenarios with a single transmitter, either single-user or multi-user. Thus, they fail when applied to UDNs since interference from neighboring cells is not considered in the design. Then, a new HB-IA is proposed for UDNs based on an iterative partial construction (IPC) procedure with the novelty of awareness of PSs quantization. The IPC algorithm relies on algebraic properties to iteratively compute the analog and digital matrices, via a partial submatrix composition, such that the Euclidean distance to the FD-IA solution is reduced. An additional digital block, based on an IA algorithm, is also applied to cancel the residual interference. Numerical results reveal that this proposal, referred as HB IPC-IA, significantly outperforms reported works achieving a near FD-IA performance. The obtained results provide practical insights for HB design in UDNs.