‘Tanalised’ timber is simply wood that’s been pressure treated with a preservative. The main problems you’ll have is that the solvents from the preservative used may react with any subsequent coats of paint. This may mean the paint doesn’t cure properly or it may discolour.
It really depends on how long the timber has been exposed to the elements for. I wouldn’t attempt to paint any wood that has been treated recently but if it has weathered for a reasonable period of time the risk of any problems will be significantly less.
First step would be to try a test area with your primer of choice (a water based quick-drying primer is probably your safest bet) and, if there are no problems, carry on as normal. If you do have problems try a solvent based primer instead.
The most likely outcome is that after priming with a water based paint it will dry OK but you’ll get patches of staining. If this is the case just use an oil based undercoat for your next coat and this should stop the stains migrating any further.