[en] Extensive first-principles calculations suggest that inverse Heusler compounds $\mathrm{M}{\mathrm{n}}_2\mathrm{S}\mathrm{i}$, $\mathrm{C}{\mathrm{r}}_2\mathrm{Z}\mathrm{n}\mathrm{G}\mathrm{e}$, $\mathrm{C}{\mathrm{r}}_2\mathrm{Z}\mathrm{n}\mathrm{S}\mathrm{n}$, $\mathrm{T}{\mathrm{i}}_2\mathrm{V}\mathrm{P}$ and $\mathrm{T}{\mathrm{i}}_2\mathrm{V}\mathrm{S}{\mathrm{b}}_{0.5}\mathrm{A}{\mathrm{s}}_{0.5}$ are the candidates to achieve fully compensated ferrimagnetic spin gapless semiconductors. It is shown that only the holes can be 100% spin polarized in $\mathrm{M}{\mathrm{n}}_2\mathrm{S}\mathrm{i}$, while both the excited electrons and the holes around the Fermi level 100% spin polarized in the others. A simple rule for searching potential fully compensated ferrimagnetic spin gapless semiconductors in Heusler compounds is proposed. Due to the spin gapless semiconducting and the fully compensated ferrimagnetic properties, these compounds offer distinct advantage towards the development of the practical spintronic devices. (letter)