The prevalence of spatially referenced multivariate data has impelled researchers to develop procedures for joint modeling of multiple spatial processes. This ordinarily involves modeling marginal and cross-process dependence for any arbitrary pair of locations using a multivariate spatial covariance function. However, building a flexible multivariate spatial covariance function that is nonnegative definite is challenging. Here, we propose a semiparametric approach for multivariate spatial covariance function estimation with approximate Matérn marginals and highly flexible cross-covariance functions via their spectral representations. The flexibility in our cross-covariance function arises due to B-spline based specification of the underlying coherence functions, which in turn allows us to capture non-trivial cross-spectral features. We then develop a likelihood-based estimation procedure and perform multiple simulation studies to demonstrate the performance of our method, especially on the coherence function estimation. Finally, we analyze particulate matter concentrations (PM2.5) and wind speed data over the West-North-Central climatic region of the United States, where we illustrate that our proposed method outperforms the commonly used full bivariate Matérn model and the linear model of coregionalization for spatial prediction.