Hi there, you are through to Dr Linda.
I'm sorry to hear Piggy is having this issue, which has certainly been going on for quite a long time.
There are a number of things that this could indicate and in cats it can be rather tricky (even for vets!) to distinguish vomiting, gagging, retching and coughing from each other.
If there are no other symptoms (such as unexplained weight loss, actual vomiting, runny stool constipation or reduced appetite) the first thing to consider given the symptoms would be 'Bilious Vomiting Syndrome' (https://www.petmd.com/cat/conditions/digestive/how-treat-bilious-vomiting-syndrome-cats), especially as the behaviour tends to occur in the morning. We would have hoped for improvement with the antacids, but it may be that he needs a longer course. Another easy thing to try would be to split his meals into smaller meals so he 'grazes' them throughout the day. Be sure to feed him last thing at night s his stomach is less empty in the morning. I would also consider switching to a sensitivity diet such as Hill's I/D or Royal Canin Gastrointestinal.
It's reassuring that the x-rays taken were normal. Can I confirm that he has also had some basic blood, urine and faecal tests performed? This would be a first step and would certainly be advised before any more invasive investigation. If normal, we may consider e.g. an endoscopic study (small camera that is used to visualise the food-pipe and stomach). Similarly, abdominal ultrasound can be useful in cases like this.
I agree that this sounds more digestive than anything else. However, I would want to confirm that the vet has had a good listen to the heart and lungs and that both sounds completely normal. Similarly, I would want to ensure that Piggy's resting breathing rate (when asleep or completely relaxed) is less than 30 breaths per minute.
As a final thought, ensure there is nothing irritant in the environment such as chemical cleaners he may be eating or cooking fumes/smoke/air fresheners he may be inhaling.
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