We report here the results obtained from a systematic optical photometric survey aimed at finding new compact binary millisecond pulsars (also known as "spiders"): the COmpact BInary PULsar SEarch. We acquired multiband optical images over 1 yr around 33 unidentified Fermi Large Area Telescope sources, selected as pulsar candidates based on their curved GeV spectra and steady γ-ray emission. We present the discovery of four optical variables coinciding with the Fermi sources 3FGL J0737.2‑3233, 3FGL J2117.6+3725 (two systems in this field), and 3FGL J2221.6+6507, which we propose as new candidate spider systems. Indeed, they all show optical flux modulation consistent with orbital periods of 0.3548(5), 0.25328(6), 0.441961(2), and 0.165(4) days, respectively, with amplitudes ≳0.3 mag and colors compatible with companion star temperatures of 5000–6000 K. These properties are consistent with the "redback" subclass of spider pulsars. If confirmed as a millisecond pulsar, 3FGL J0737.2‑3233 will be the closest known spider to Earth ( , from Gaia-DR3 parallax). We searched and did not find any X-ray sources matching our four candidates, placing 3σ upper limits of ∼1031–1032 erg s‑1 (0.3–10 keV) on their soft X-ray luminosities. We also present and discuss other multiwavelength information on our spider candidates, from infrared to X-rays.