Alkali based mortars and cements
If we ignore CaCO3 altogether:
Yes,using aluminosilicate geopolymers/alkali activated cements.
You can make cements using sodium or potassium hydroxide, and an amorphous aluminosilicate mix, such as certain pozzolanic soils, flyash, dehydrated clays, geothermal silica, etc. Obviously, it'll be much more costly than Portland cement or lime mortar as alkali hydroxides are comparatively scarce. Nonetheless, it's possible.
If you had adjacent deposits of natron and kaolinite, you'd be well on your way. It could be discovered as a side product of a porcelain making operation.
Another alternative for indoors only is bonding things with sodium silicate (waterglass), which you can make from sodium hydroxide and geothermal silica. You just mix up any powder in it and it sets. It is insanely, incredibly hard and strong...BUT it is attacked by water, so can only be used in a completely dry environment, which is why it's not used everywhere.