The correlations between photons generated by nonlinear optical processes offer advantages for many quantum technology applications, including spectroscopy, imaging, and metrology. Here, we use spontaneous four-wave mixing in a birefringent single-mode fiber pumped by a tunable pulsed laser as a broadly tunable source of phase-matched non-degenerate photon pairs for spectroscopy. The pairs are tunable such that the idler beam measures the transmittance spectrum of a sample in the near infrared, while the visible signal beam independently reports correlation information. By the time-resolved counting of both signal and idler photons, we use photon-number correlations to remove uncorrelated noise from the probe beam. Here, we have used heralded spectroscopy to measure the absorption spectrum of gallium arsenide near its band edge, despite the idler photon spectrum being dominated by a large background from spontaneous Raman scattering.