A new population of neutron stars has emerged during the last decade: compact binary millisecond pulsars (CBMSPs). Because these pulsars and their companion stars are in tight orbits with typical separations of $10^{11}$ cm, their winds interact strongly forming an intrabinary shock. Electron-positron pairs reaccelerated at the shock can reach energies of about 10 TeV, which makes this new population a potential source of GeV-TeV cosmic ray positrons. We present an analytical model for the fluxes and spectra of positrons from intrabinary shocks of CBMSPs. We find that the minimum energy $E_{min}$ of the pairs that enter the shock is critical to quantify the energy spectrum with which positrons are injected into the interstellar medium. We measure for the first time the Galactic scale height of CBMSPs, $z_e=0.4pm0.1$ kpc, after correcting for an observational bias against finding them close to the Galactic plane. From this, we estimate a local density of 5-9 kpc$^{-3}$ and an extrapolated total of 2-7 thousand CBMSPs in the Galaxy. We then propagate the pairs in the isotropic diffusion approximation and find that the positron flux from the total population is about two times higher than that from the 52 currently known systems. For $E_{min}$ between 1 and 50 GeV, our model predicts only a minor contribution from CBMSPs to the diffuse positron flux at 100 GeV observed at Earth. We also quantify the effects of anisotropic transport due to the ordered Galactic magnetic field, which can change the diffuse flux from nearby sources drastically. Finally, we find that a single hidden CBMSP close to the Galactic plane can yield a positron flux comparable to the AMS-02 measurements at 600 GeV if its line-of-sight to Earth is along the ordered Galactic field lines, while its combined electron and positron flux at higher energies would be close to the measurements of CALET, DAMPE and Fermi-LAT.
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