A friend is in a similar position. He is of the opinion that the bank cannot look for more than what they will get when they sell your Portuguese property. It's similar in the US I believe. In other words your indebtedness to the bank is limited to what they can get when they sell your property. After they sell your property they can keep all they get up to what you owe them and then that's it.
That does not sound logical or reasonable. The OP owes the bank an quantifiable sum which they are pursuing, "what the bank would get for selling your Portuguese property" is not quantifiable until a sale is agreed. On what does your friend base his opinion? Does he have something more concrete to back that position?
OP, from reading through this thread, the answer to your question seems to be yes, they can pursue you in Ireland for the debt through the same means available if you owed the money in Ireland. If they get a judgement order it can be enforced like any other judgement order. The order will exist for 12 years and can be converted into a judgement mortgage against your property. This can be paid from the proceeds of the sale of your home only after your home mortgage is paid off. That judgement mortgage will only be of value to them if you have positive equity in your property. If you do, they can try to force the sale of your home to recover the money. You have the same defensive options open to you as you would if they were Irish.
You can advise them and perhaps demonsrate that there is nothing to be gained for them but that will not prevent them trying.
I think you need to stop seeing it as a "Portuguese" debt. It reads almost as if you hoped the different jurisdiction would be a protection and a distance, a firewall between you and there. That trouble in one jurisdiction was safely confined there.
You need to sit down and lay out your options in terms of your property and debt as a whole and not try to view any one part of it in isolation. You need to try and get a reasonable idea of the value of your Irish property to assess the risk to your home (300k - 500k is a very wide margin of error!). You need to investigate the possibility of selling your Portuguese property, what would be the shortfall, whether there is any option of converting the NE into unsecured debt at a reasonable rate and whether you could afford such a solution.
It won't be a simple or a comfortable exercise but at least it will give you a more secure position to argue any case from.