(Abridged) Any viable cosmological model in which galaxies interact predicts the existence of primordial and tidal dwarf galaxies (TDGs). In particular, in the standard model of cosmology ($Lambda$CDM), according to the dual dwarf galaxy theorem, there must exist both primordial dark matter-dominated and dark matter-free TDGs with different radii. We study the frequency, evolution, and properties of TDGs in a $Lambda$CDM cosmology. We use the hydrodynamical cosmological Illustris-1 simulation to identify tidal dwarf galaxy candidates (TDGCs) and study their present-day physical properties. We also present movies on the formation of a few galaxies lacking dark matter, confirming their tidal dwarf nature. TDGCs can however also be formed via other mechanisms, such as from ram-pressure-stripped material or, speculatively, from cold-accreted gas. We find 97 TDGCs with $M_{stellar} >5 times 10^7 M_odot$ at redshift $z = 0$, corresponding to a co-moving number density of $2.3 times 10^{-4} h^3 cMpc^{-3}$. The most massive TDGC has $M_{total} = 3.1 times 10^9 M_odot$, comparable to that of the Large Magellanic Cloud. TDGCs are phase-space-correlated, reach high metallicities, and are typically younger than dark matter-rich dwarf galaxies. We report for the first time the verification of the dual dwarf theorem in a self-consistent $Lambda$CDM cosmological simulation. Simulated TDGCs and dark matter-dominated galaxies populate different regions in the radius-mass diagram in disagreement with observations of early-type galaxies. The dark matter-poor galaxies formed in Illustris-1 have comparable radii to observed dwarf galaxies and to TDGs formed in other galaxy-encounter simulations. In Illustris-1, only 0.17% of all selected galaxies with $M_{stellar} = 5 times 10^7-10^9 M_odot$ are TDGCs or dark matter-poor dwarf galaxies. The occurrence of NGC 1052-DF2-type objects is discussed.
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