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There isn't an abundance of Carp fishing locally, however there are a couple of local places near to you.
The first is El Bosquet, which is in an area called 'the fondo' a quick search on the title will give you the directions and layout. It's a purpose built system of canals/drains well stocked and regularly fished by local angling societies. It isn't the prettiest but is somewhere you will catch plenty of fish. From memory I think it's 7 euro's a day.
The second is, as you come along the 332 out of Guardamar towards Torrevieja you pass the roundabout for the CV 895 (lemon tree road) the next roundabout on your right hand side you will see a restaurant (cant remember the name) It's a popular bowling club. Anyway go in there and there is a small pool attached to it, it is small but does hold some fish. From memory they used to sell a day ticket with a breakfast included.
I would recommend a spin out to both for a recce and you will see people there fishing and as I am sure you know aways ready to chat !
You will see a lot of locals parked near to the drains/water channels all over, as these do hold carp.
There may be other places but I haven't found them yet. If you look a bit further afield, Valencia has some very good fishing spots, specifically 'the river Ebro' which is a catfish and carp Mecca.
Most importantly you will need a licence to fish which tackle shops can assist you with or you can do it yourself.
There are two permits available, one for fresh water and one for sea fishing. Both available in Alicante to do yourself, you need ID, and it's a bit of a pain but not too difficult, you have to complete the forms, go to the local bank, pay the sums then return for the completion of paperwork and the licence is then issued. Probably an hours job.
Alternatively some of the local tackle shops will do it for you at a charge. Easy enough to DIY.
Ministerio de Medio Ambiente
At: Calle Churruca No29, AlicanteThis is about 400 meters inland from the new bus station in Alicante
Tel: 965 148 111Probably your local tackle shop is La Pesca, Avenida de Palacios 21,Rojales (Formentera), they stock tackle, baits and also will process your licence.