Short answer. No, not without permission. The Civil Code bars an industrial partner from engaging in business for himself unless the partnership expressly allows it. If he does anyway, the capitalist partners may exclude him from the firm or keep the benefits he gained from it, plus damages either way.
What the law says
An industrial partner cannot engage in business for himself, unless the partnership expressly permits him to do so; and if he should do so, the capitalist partners may either exclude him from the firm or avail themselves of the benefits which he may have obtained in violation of this provision, with a right to damages in either case.
Civil Code, Article 1789 — Industrial Partner's Exclusivity. Read the full provision →
The prohibition is broader than "competing" business
The question is usually framed around competition, but Article 1789 does not use that word. It says an industrial partner cannot engage in business for himself at all, full stop, unless the partnership expressly permits it. A side venture in a completely different line of work is covered just as much as a rival firm in the same trade — what triggers the article is the industrial partner devoting himself to business of his own, not specifically whether that business steals customers from the partnership.
Why industrial partners are held to this and capitalist partners are not
An industrial partner contributes labor and skill rather than capital, and the partnership is entitled to that labor undivided. Running an outside business necessarily draws on the same time, attention, and effort the partnership is relying on, which is why the Code holds industrial partners to a stricter standard here than it holds partners who contribute capital. The exception — express permission from the partnership — exists precisely because the partners can agree to relax this if they choose to; the default assumes they have not.
Two separate remedies, and damages either way
If an industrial partner engages in outside business without permission, Article 1789 gives the capitalist partners a choice, not a single fixed consequence: they may exclude him from the firm, ending the partnership relationship with him, or they may avail themselves of the benefits he obtained from the outside venture, effectively claiming those gains for the partnership instead. Either path carries a right to damages on top of it — the choice between exclusion and claiming the benefits does not use up the partnership's separate claim for whatever loss the breach caused.
What to check before acting on this
If you are a capitalist partner discovering that an industrial partner is running an outside business, first check the partnership agreement for any clause expressly permitting outside work — the prohibition only applies in its absence. If you are the industrial partner and want to take on other work, get that permission recorded in writing rather than assuming an unspoken understanding will hold up, since the article's protection for you depends on the partnership having actually granted it.