It depends on the PSPO, HMG reckon that carrying poo bags and a scoop (I don't own one, handling my own dog's poo through a bag isn't a problem) can be required by the PSPO.
https://www.gov.uk/control-dog-public/public-spaces-protection-orders (enforcement is by £100 on the spot fines, or risk upto £1000 in Court)
My own Council (Walsall) have one PSPO, covering the town centre, and the relevant section of must dos is
It is ordered that the following specified things are done by persons carrying on specified activities in this restricted area.
1. Any person in charge of a dog in a public place must keep the dog on a lead or otherwise under physical control at all times.
I presume that they could insert a point about carrying the means to clear up any dog fouling at all times in this section, and anything else that isn't discriminatory that they feel like- wearing an upper body covering (as well as a lower body covering) springs to mind.
The BBC reported that someone was fined for no poo bags
https://www.bbc.co.uk/news/articles/c4gpzxnl7l9oI recall reading a thread somewhere about an owner being fined because they had used their last bag clearing up and no longer had the means to clear up as required, but I can't track it down.
EDIT:
I don't know why I didn't think of this in the first place, but Manchester publish their PSPOs
https://www.manchester.gov.uk/directory_record/438586/dog_fouling/category/1732/control_of_dogsand indeed it does state
A person in charge of a dog on land specified in the Schedule to this Order must have with him an appropriate means to pick up dog faeces deposited by that dog forthwith unless :-
(a) he has reasonable excuse for failing to do so; or
(b) the owner, occupier or other person or authority having control of the land has consented (generally or specifically) to his failing to do so.
The obligation is complied with if, after a request from an authorised officer or constable, the person in charge of the dog produces an appropriate means to pick up dog faeces.