  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, $z_e=0.4\pm0.1$ 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.
