This paper is the 4th in a series describing the latest additions to the BaSTI stellar evolution database, which consists of a large set of homogeneous models and tools for population synthesis studies. Here we present a new set of low and high resolution synthetic spectra based on the BaSTI stellar models, covering a large range of simple stellar populations (SSPs) for both scaled solar and alpha-enhanced metal mixtures. This enables a completely consistent study of the photometric and spectroscopic properties of both resolved and unresolved stellar populations, and allows us to make detailed tests on their integrated properties. Our low resolution spectra are suitable for deriving broadband magnitudes and colors in any photometric system. These spectra cover the full wavelength range (9-160000nm) and include all evolutionary stages up to the end of AGB evolution. Our high resolution spectra are suitable for studying the behaviour of line indices and we have tested them against a large sample of Galactic globular clusters. We find that the range of ages, iron abundances [Fe/H], and degree of alpha-enhancement predicted by the models matches observed values very well. We have also tested the global consistency of the BaSTI models by making detailed comparisons between ages and metallicities derived from isochrone fitting to observed CMDs, and from line index strengths, for the Galactic globular cluster 47Tuc and the open cluster M67. For 47Tuc we find reasonable agreement between the 2 methods, within the estimated errors. From the comparison with M67 we find non-negligible effects on derived line indices caused by statistical fluctuations, which are a result of the specific method used to populate an isochrone and assign appropriate spectra to individual stars. (abridged)
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