The calibration and analysis of polarization observations in very long baseline interferometry (VLBI) requires the use of specific algorithms that suffer from several limitations, closely related to assumptions in the data properties that may not hold in observations taken with new-generation VLBI equipment. Currently, the instantaneous bandwidth achievable with VLBI backends can be as high as several gigahertz, covering several radio bands simultaneously. In addition, the sensitivity of VLBI observations with the most updated equipment may reach dynamic ranges of tens of thousands, both in total intensity and polarization. In this paper, we discuss the impact of the limitations of common VLBI polarimetry algorithms on narrow-field observations taken with modern VLBI arrays, from the VLBI Global Observing System to the Event Horizon Telescope, and present new software that overcomes these limitations. In particular, our software is able to perform a simultaneous fit of multiple calibrator sources, include nonlinear terms in the model of the instrumental polarization and use a self-calibration approach for the estimate of the polarization leakage in the antenna receivers.