Place of supply - reg

As per section 12(3) of the IGST act 2017 ,place of supply in case of Immoveable is where the property located , so place of supply will be Tamil Nadu . In my opinion you can raised IGST tax invoice . please confirm with others also .

No....
Section 22 Of The ACT Reads As 'Every Supplier Shall Be Liable To Be Registered Under This Act In The State Or Union Territory, Other Than Special Category States, From Where He Makes A Taxable Supply Of Goods Or Services Or Both.

In above WHERE means where he is registered.
So if you are registered Karnataka you can supply services from Karnataka to Tamil Nadu where the property is situated.

Now as Prasad Ji has quoted the the place of supply in term of Section 12(3) IGST Act , will the location of property situated, hence you can raise invoice by charging IGST on bill for rental from Tamil Nadu. No need to take separate registration in Tamil Nadu.

Note : In case of any inward supply in regard to properly in Tamil Nadu ,you won't able to avail ITC on same as the POS WILL BE THE Tamil Nadu .

*can raise invoice from Karnataka to Tamil Nadu by charging IGST

Agree with Pankaj ji